Article PARTFUN2, MML version 4.99.1005
:: PARTFUN2:th 3
theorem
for b1, b2 being non empty set
for b3, b4 being Function-like Relation of b2,b1
st dom b3 = dom b4 &
(for b5 being Element of b2
st b5 in dom b3
holds b3 /. b5 = b4 /. b5)
holds b3 = b4;
:: PARTFUN2:th 4
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Function-like Relation of b3,b2 holds
b1 in rng b4
iff
ex b5 being Element of b3 st
b5 in dom b4 & b1 = b4 /. b5;
:: PARTFUN2:th 6
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like Relation of b3,b1
for b5 being Function-like Relation of b1,b2
for b6 being Function-like Relation of b3,b2 holds
b6 = b5 * b4
iff
(for b7 being Element of b3 holds
b7 in dom b6
iff
b7 in dom b4 & b4 /. b7 in dom b5) &
(for b7 being Element of b3
st b7 in dom b6
holds b6 /. b7 = b5 /. (b4 /. b7));
:: PARTFUN2:th 9
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Function-like Relation of b1,b2
for b6 being Function-like Relation of b2,b3
st b4 in dom b5 & b5 /. b4 in dom b6
holds (b6 * b5) /. b4 = b6 /. (b5 /. b4);
:: PARTFUN2:th 10
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Function-like Relation of b1,b2
for b6 being Function-like Relation of b2,b3
st rng b5 c= dom b6 & b4 in dom b5
holds (b6 * b5) /. b4 = b6 /. (b5 /. b4);
:: PARTFUN2:funcnot 1 => PARTFUN2:func 1
definition
let a1 be non empty set;
let a2 be Element of bool a1;
redefine func id a2 -> Function-like Relation of a1,a1;
end;
:: PARTFUN2:th 12
theorem
for b1 being non empty set
for b2 being Element of bool b1
for b3 being Function-like Relation of b1,b1 holds
b3 = id b2
iff
dom b3 = b2 &
(for b4 being Element of b1
st b4 in b2
holds b3 /. b4 = b4);
:: PARTFUN2:th 14
theorem
for b1 being non empty set
for b2 being Element of bool b1
for b3 being Element of b1
for b4 being Function-like Relation of b1,b1
st b3 in (dom b4) /\ b2
holds b4 /. b3 = (b4 * id b2) /. b3;
:: PARTFUN2:th 15
theorem
for b1 being non empty set
for b2 being Element of bool b1
for b3 being Element of b1
for b4 being Function-like Relation of b1,b1 holds
b3 in dom ((id b2) * b4)
iff
b3 in dom b4 & b4 /. b3 in b2;
:: PARTFUN2:th 16
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b2,b1
st for b4, b5 being Element of b2
st b4 in dom b3 & b5 in dom b3 & b3 /. b4 = b3 /. b5
holds b4 = b5
holds b3 is one-to-one;
:: PARTFUN2:th 17
theorem
for b1, b2 being set
for b3, b4 being non empty set
for b5 being Function-like Relation of b3,b4
st b5 is one-to-one & b1 in dom b5 & b2 in dom b5 & b5 /. b1 = b5 /. b2
holds b1 = b2;
:: PARTFUN2:funcreg 1
registration
cluster {} -> one-to-one;
end;
:: PARTFUN2:exreg 1
registration
let a1, a2 be set;
cluster Relation-like Function-like one-to-one Relation of a1,a2;
end;
:: PARTFUN2:funcnot 2 => PARTFUN2:func 2
definition
let a1, a2 be set;
let a3 be Function-like one-to-one Relation of a1,a2;
redefine func a3 " -> Function-like Relation of a2,a1;
end;
:: PARTFUN2:th 18
theorem
for b1, b2 being non empty set
for b3 being Function-like one-to-one Relation of b1,b2
for b4 being Function-like Relation of b2,b1 holds
b4 = b3 "
iff
dom b4 = rng b3 &
(for b5 being Element of b2
for b6 being Element of b1 holds
b5 in rng b3 & b6 = b4 /. b5
iff
b6 in dom b3 & b5 = b3 /. b6);
:: PARTFUN2:th 22
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Function-like one-to-one Relation of b1,b2
st b3 in dom b4
holds b3 = b4 " /. (b4 /. b3) & b3 = (b4 " * b4) /. b3;
:: PARTFUN2:th 23
theorem
for b1, b2 being non empty set
for b3 being Element of b2
for b4 being Function-like one-to-one Relation of b1,b2
st b3 in rng b4
holds b3 = b4 /. (b4 " /. b3) & b3 = (b4 * (b4 ")) /. b3;
:: PARTFUN2:th 24
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,b2
for b4 being Function-like Relation of b2,b1
st b3 is one-to-one &
dom b3 = rng b4 &
rng b3 = dom b4 &
(for b5 being Element of b1
for b6 being Element of b2
st b5 in dom b3 & b6 in dom b4
holds b3 /. b5 = b6
iff
b4 /. b6 = b5)
holds b4 = b3 ";
:: PARTFUN2:th 32
theorem
for b1 being set
for b2, b3 being non empty set
for b4, b5 being Function-like Relation of b3,b2 holds
b4 = b5 | b1
iff
dom b4 = (dom b5) /\ b1 &
(for b6 being Element of b3
st b6 in dom b4
holds b4 /. b6 = b5 /. b6);
:: PARTFUN2:th 34
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of b2
for b5 being Function-like Relation of b2,b3
st b4 in (dom b5) /\ b1
holds (b5 | b1) /. b4 = b5 /. b4;
:: PARTFUN2:th 35
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of b2
for b5 being Function-like Relation of b2,b3
st b4 in dom b5 & b4 in b1
holds (b5 | b1) /. b4 = b5 /. b4;
:: PARTFUN2:th 36
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of b2
for b5 being Function-like Relation of b2,b3
st b4 in dom b5 & b4 in b1
holds b5 /. b4 in rng (b5 | b1);
:: PARTFUN2:funcnot 3 => PARTFUN2:func 3
definition
let a1, a2 be non empty set;
let a3 be set;
let a4 be Function-like Relation of a1,a2;
redefine func a3 | a4 -> Function-like Relation of a1,a2;
end;
:: PARTFUN2:th 37
theorem
for b1 being set
for b2, b3 being non empty set
for b4, b5 being Function-like Relation of b3,b2 holds
b4 = b1 | b5
iff
(for b6 being Element of b3 holds
b6 in dom b4
iff
b6 in dom b5 & b5 /. b6 in b1) &
(for b6 being Element of b3
st b6 in dom b4
holds b4 /. b6 = b5 /. b6);
:: PARTFUN2:th 38
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of b2
for b5 being Function-like Relation of b2,b3 holds
b4 in dom (b1 | b5)
iff
b4 in dom b5 & b5 /. b4 in b1;
:: PARTFUN2:th 39
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of b2
for b5 being Function-like Relation of b2,b3
st b4 in dom (b1 | b5)
holds (b1 | b5) /. b4 = b5 /. b4;
:: PARTFUN2:th 40
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of bool b2
for b5 being Function-like Relation of b3,b2 holds
b4 = b5 .: b1
iff
for b6 being Element of b2 holds
b6 in b4
iff
ex b7 being Element of b3 st
b7 in dom b5 & b7 in b1 & b6 = b5 /. b7;
:: PARTFUN2:th 41
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of b3
for b5 being Function-like Relation of b2,b3 holds
b4 in b5 .: b1
iff
ex b6 being Element of b2 st
b6 in dom b5 & b6 in b1 & b4 = b5 /. b6;
:: PARTFUN2:th 42
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Function-like Relation of b1,b2
st b3 in dom b4
holds Im(b4,b3) = {b4 /. b3};
:: PARTFUN2:th 43
theorem
for b1, b2 being non empty set
for b3, b4 being Element of b1
for b5 being Function-like Relation of b1,b2
st b3 in dom b5 & b4 in dom b5
holds b5 .: {b3,b4} = {b5 /. b3,b5 /. b4};
:: PARTFUN2:th 44
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of bool b3
for b5 being Function-like Relation of b3,b2 holds
b4 = b5 " b1
iff
for b6 being Element of b3 holds
b6 in b4
iff
b6 in dom b5 & b5 /. b6 in b1;
:: PARTFUN2:th 46
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b2,b1 holds
ex b4 being Function-like quasi_total Relation of b2,b1 st
for b5 being Element of b2
st b5 in dom b3
holds b4 . b5 = b3 /. b5;
:: PARTFUN2:th 47
theorem
for b1, b2 being non empty set
for b3, b4 being Function-like Relation of b2,b1 holds
b3 tolerates b4
iff
for b5 being Element of b2
st b5 in (dom b3) /\ dom b4
holds b3 /. b5 = b4 /. b5;
:: PARTFUN2:sch 1
scheme PARTFUN2:sch 1
{F1 -> non empty set,
F2 -> non empty set}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being Element of F1() holds
b2 in dom b1
iff
ex b3 being Element of F2() st
P1[b2, b3]) &
(for b2 being Element of F1()
st b2 in dom b1
holds P1[b2, b1 /. b2])
:: PARTFUN2:sch 2
scheme PARTFUN2:sch 2
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2()}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being Element of F1() holds
b2 in dom b1
iff
P1[b2]) &
(for b2 being Element of F1()
st b2 in dom b1
holds b1 /. b2 = F3(b2))
:: PARTFUN2:sch 3
scheme PARTFUN2:sch 3
{F1 -> non empty set,
F2 -> non empty set,
F3 -> set,
F4 -> Element of F2()}:
for b1, b2 being Function-like Relation of F1(),F2()
st dom b1 = F3() &
(for b3 being Element of F1()
st b3 in dom b1
holds b1 /. b3 = F4(b3)) &
dom b2 = F3() &
(for b3 being Element of F1()
st b3 in dom b2
holds b2 /. b3 = F4(b3))
holds b1 = b2
:: PARTFUN2:funcnot 4 => PARTFUN2:func 4
definition
let a1, a2 be non empty set;
let a3 be Element of bool a1;
let a4 be Element of a2;
redefine func a3 --> a4 -> Function-like Relation of a1,a2;
end;
:: PARTFUN2:th 48
theorem
for b1, b2 being non empty set
for b3 being Element of bool b1
for b4 being Element of b1
for b5 being Element of b2
st b4 in b3
holds (b3 --> b5) /. b4 = b5;
:: PARTFUN2:th 49
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Function-like Relation of b2,b1
st for b5 being Element of b2
st b5 in dom b4
holds b4 /. b5 = b3
holds b4 = (dom b4) --> b3;
:: PARTFUN2:th 50
theorem
for b1, b2, b3 being non empty set
for b4 being Element of bool b1
for b5 being Element of b2
for b6 being Function-like Relation of b2,b3
st b5 in dom b6
holds b6 * (b4 --> b5) = b4 --> (b6 /. b5);
:: PARTFUN2:th 51
theorem
for b1 being non empty set
for b2 being Element of bool b1 holds
id b2 is total(b1, b1)
iff
b2 = b1;
:: PARTFUN2:th 52
theorem
for b1, b2 being non empty set
for b3 being Element of bool b1
for b4 being Element of b2
st b3 --> b4 is total(b1, b2)
holds b3 <> {};
:: PARTFUN2:th 53
theorem
for b1, b2 being non empty set
for b3 being Element of bool b2
for b4 being Element of b1 holds
b3 --> b4 is total(b2, b1)
iff
b3 = b2;
:: PARTFUN2:prednot 1 => PARTFUN2:pred 1
definition
let a1, a2 be non empty set;
let a3 be Function-like Relation of a1,a2;
let a4 be set;
pred A3 is_constant_on A4 means
ex b1 being Element of a2 st
for b2 being Element of a1
st b2 in a4 /\ dom a3
holds a3 /. b2 = b1;
end;
:: PARTFUN2:dfs 1
definiens
let a1, a2 be non empty set;
let a3 be Function-like Relation of a1,a2;
let a4 be set;
To prove
a3 is_constant_on a4
it is sufficient to prove
thus ex b1 being Element of a2 st
for b2 being Element of a1
st b2 in a4 /\ dom a3
holds a3 /. b2 = b1;
:: PARTFUN2:def 3
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,b2
for b4 being set holds
b3 is_constant_on b4
iff
ex b5 being Element of b2 st
for b6 being Element of b1
st b6 in b4 /\ dom b3
holds b3 /. b6 = b5;
:: PARTFUN2:th 55
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Function-like Relation of b3,b2 holds
b4 is_constant_on b1
iff
for b5, b6 being Element of b3
st b5 in b1 /\ dom b4 & b6 in b1 /\ dom b4
holds b4 /. b5 = b4 /. b6;
:: PARTFUN2:th 56
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Function-like Relation of b2,b3
st b1 meets dom b4
holds b4 is_constant_on b1
iff
ex b5 being Element of b3 st
rng (b4 | b1) = {b5};
:: PARTFUN2:th 57
theorem
for b1, b2 being set
for b3, b4 being non empty set
for b5 being Function-like Relation of b3,b4
st b5 is_constant_on b1 & b2 c= b1
holds b5 is_constant_on b2;
:: PARTFUN2:th 58
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Function-like Relation of b2,b3
st b1 misses dom b4
holds b4 is_constant_on b1;
:: PARTFUN2:th 59
theorem
for b1, b2 being non empty set
for b3 being Element of bool b1
for b4 being Element of b2
for b5 being Function-like Relation of b1,b2
st b5 | b3 = (dom (b5 | b3)) --> b4
holds b5 is_constant_on b3;
:: PARTFUN2:th 60
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Function-like Relation of b2,b3 holds
b4 is_constant_on {b1};
:: PARTFUN2:th 61
theorem
for b1, b2 being set
for b3, b4 being non empty set
for b5 being Function-like Relation of b3,b4
st b5 is_constant_on b1 & b5 is_constant_on b2 & b1 /\ b2 meets dom b5
holds b5 is_constant_on b1 \/ b2;
:: PARTFUN2:th 62
theorem
for b1, b2 being set
for b3, b4 being non empty set
for b5 being Function-like Relation of b3,b4
st b5 is_constant_on b1
holds b5 | b2 is_constant_on b1;
:: PARTFUN2:th 63
theorem
for b1, b2 being non empty set
for b3 being Element of bool b1
for b4 being Element of b2 holds
b3 --> b4 is_constant_on b3;
:: PARTFUN2:th 64
theorem
for b1, b2 being non empty set
for b3, b4 being Function-like Relation of b2,b1 holds
b3 c= b4
iff
dom b3 c= dom b4 &
(for b5 being Element of b2
st b5 in dom b3
holds b3 /. b5 = b4 /. b5);
:: PARTFUN2:th 65
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Element of b2
for b5 being Function-like Relation of b1,b2 holds
b3 in dom b5 & b4 = b5 /. b3
iff
[b3,b4] in b5;
:: PARTFUN2:th 66
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b1
for b5 being Element of b3
for b6 being Function-like Relation of b1,b2
for b7 being Function-like Relation of b2,b3
st [b4,b5] in b7 * b6
holds [b4,b6 /. b4] in b6 & [b6 /. b4,b5] in b7;
:: PARTFUN2:th 67
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Element of b2
for b5 being Function-like Relation of b1,b2
st b5 = {[b3,b4]}
holds b5 /. b3 = b4;
:: PARTFUN2:th 68
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Function-like Relation of b1,b2
st dom b4 = {b3}
holds b4 = {[b3,b4 /. b3]};
:: PARTFUN2:th 69
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4, b5, b6 being Function-like Relation of b1,b2
st b4 = b5 /\ b6 & b3 in dom b4
holds b4 /. b3 = b5 /. b3 & b4 /. b3 = b6 /. b3;
:: PARTFUN2:th 70
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4, b5, b6 being Function-like Relation of b1,b2
st b3 in dom b4 & b5 = b4 \/ b6
holds b5 /. b3 = b4 /. b3;
:: PARTFUN2:th 71
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4, b5, b6 being Function-like Relation of b1,b2
st b3 in dom b4 & b5 = b6 \/ b4
holds b5 /. b3 = b4 /. b3;
:: PARTFUN2:th 72
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4, b5, b6 being Function-like Relation of b1,b2
st b3 in dom b4 & b4 = b5 \/ b6 & b4 /. b3 <> b5 /. b3
holds b4 /. b3 = b6 /. b3;
:: PARTFUN2:th 73
theorem
for b1, b2 being non empty set
for b3 being Element of bool b1
for b4 being Element of b1
for b5 being Function-like Relation of b1,b2 holds
b4 in dom b5 & b4 in b3
iff
[b4,b5 /. b4] in b5 | b3;
:: PARTFUN2:th 74
theorem
for b1, b2 being non empty set
for b3 being Element of bool b2
for b4 being Element of b1
for b5 being Function-like Relation of b1,b2 holds
b4 in dom b5 & b5 /. b4 in b3
iff
[b4,b5 /. b4] in b3 | b5;
:: PARTFUN2:th 75
theorem
for b1, b2 being non empty set
for b3 being Element of bool b2
for b4 being Element of b1
for b5 being Function-like Relation of b1,b2 holds
b4 in b5 " b3
iff
[b4,b5 /. b4] in b5 & b5 /. b4 in b3;
:: PARTFUN2:th 76
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Function-like Relation of b3,b2 holds
b4 is_constant_on b1
iff
ex b5 being Element of b2 st
for b6 being Element of b3
st b6 in b1 /\ dom b4
holds b4 . b6 = b5;
:: PARTFUN2:th 77
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Function-like Relation of b3,b2 holds
b4 is_constant_on b1
iff
for b5, b6 being Element of b3
st b5 in b1 /\ dom b4 & b6 in b1 /\ dom b4
holds b4 . b5 = b4 . b6;
:: PARTFUN2:th 78
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Element of b2
for b5 being Function-like Relation of b3,b2
st b4 in b5 .: b1
holds ex b6 being Element of b3 st
b6 in dom b5 & b6 in b1 & b4 = b5 . b6;
:: PARTFUN2:th 79
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Element of b2
for b5 being Function-like Relation of b1,b2
st b5 is one-to-one
holds b4 in rng b5 & b3 = b5 " . b4
iff
b3 in dom b5 & b4 = b5 . b3;