Article ZF_COLLA, MML version 4.99.1005
:: ZF_COLLA:funcnot 1 => ZF_COLLA:func 1
definition
let a1 be non empty set;
let a2 be ordinal set;
func Collapse(A1,A2) -> set means
ex b1 being Relation-like Function-like T-Sequence-like set st
it = {b2 where b2 is Element of a1: for b3 being Element of a1
st b3 in b2
holds ex b4 being ordinal set st
b4 in proj1 b1 & b3 in union {b1 . b4}} &
proj1 b1 = a2 &
(for b2 being ordinal set
st b2 in a2
holds b1 . b2 = {b3 where b3 is Element of a1: for b4 being Element of a1
st b4 in b3
holds ex b5 being ordinal set st
b5 in proj1 (b1 | b2) &
b4 in union {(b1 | b2) . b5}});
end;
:: ZF_COLLA:def 1
theorem
for b1 being non empty set
for b2 being ordinal set
for b3 being set holds
b3 = Collapse(b1,b2)
iff
ex b4 being Relation-like Function-like T-Sequence-like set st
b3 = {b5 where b5 is Element of b1: for b6 being Element of b1
st b6 in b5
holds ex b7 being ordinal set st
b7 in proj1 b4 & b6 in union {b4 . b7}} &
proj1 b4 = b2 &
(for b5 being ordinal set
st b5 in b2
holds b4 . b5 = {b6 where b6 is Element of b1: for b7 being Element of b1
st b7 in b6
holds ex b8 being ordinal set st
b8 in proj1 (b4 | b5) &
b7 in union {(b4 | b5) . b8}});
:: ZF_COLLA:th 1
theorem
for b1 being non empty set
for b2 being ordinal set holds
Collapse(b1,b2) = {b3 where b3 is Element of b1: for b4 being Element of b1
st b4 in b3
holds ex b5 being ordinal set st
b5 in b2 & b4 in Collapse(b1,b5)};
:: ZF_COLLA:th 2
theorem
for b1 being non empty set
for b2 being Element of b1 holds
for b3 being Element of b1 holds
not b3 in b2
iff
b2 in Collapse(b1,{});
:: ZF_COLLA:th 3
theorem
for b1 being non empty set
for b2 being ordinal set
for b3 being Element of b1 holds
b3 /\ b1 c= Collapse(b1,b2)
iff
b3 in Collapse(b1,succ b2);
:: ZF_COLLA:th 4
theorem
for b1 being non empty set
for b2, b3 being ordinal set
st b2 c= b3
holds Collapse(b1,b2) c= Collapse(b1,b3);
:: ZF_COLLA:th 5
theorem
for b1 being non empty set
for b2 being Element of b1 holds
ex b3 being ordinal set st
b2 in Collapse(b1,b3);
:: ZF_COLLA:th 6
theorem
for b1 being non empty set
for b2 being ordinal set
for b3, b4 being Element of b1
st b3 in b4 & b4 in Collapse(b1,b2)
holds b3 in Collapse(b1,b2) &
(ex b5 being ordinal set st
b5 in b2 & b3 in Collapse(b1,b5));
:: ZF_COLLA:th 7
theorem
for b1 being non empty set
for b2 being ordinal set holds
Collapse(b1,b2) c= b1;
:: ZF_COLLA:th 8
theorem
for b1 being non empty set holds
ex b2 being ordinal set st
b1 = Collapse(b1,b2);
:: ZF_COLLA:th 9
theorem
for b1 being non empty set holds
ex b2 being Relation-like Function-like set st
proj1 b2 = b1 &
(for b3 being Element of b1 holds
b2 . b3 = b2 .: b3);
:: ZF_COLLA:prednot 1 => ZF_COLLA:pred 1
definition
let a1 be Relation-like Function-like set;
let a2, a3 be set;
pred A1 is_epsilon-isomorphism_of A2,A3 means
proj1 a1 = a2 &
proj2 a1 = a3 &
a1 is one-to-one &
(for b1, b2 being set
st b1 in a2 & b2 in a2
holds ex b3 being set st
b3 = b2 & b1 in b3
iff
ex b3 being set st
a1 . b2 = b3 & a1 . b1 in b3);
end;
:: ZF_COLLA:dfs 2
definiens
let a1 be Relation-like Function-like set;
let a2, a3 be set;
To prove
a1 is_epsilon-isomorphism_of a2,a3
it is sufficient to prove
thus proj1 a1 = a2 &
proj2 a1 = a3 &
a1 is one-to-one &
(for b1, b2 being set
st b1 in a2 & b2 in a2
holds ex b3 being set st
b3 = b2 & b1 in b3
iff
ex b3 being set st
a1 . b2 = b3 & a1 . b1 in b3);
:: ZF_COLLA:def 2
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set holds
b1 is_epsilon-isomorphism_of b2,b3
iff
proj1 b1 = b2 &
proj2 b1 = b3 &
b1 is one-to-one &
(for b4, b5 being set
st b4 in b2 & b5 in b2
holds ex b6 being set st
b6 = b5 & b4 in b6
iff
ex b6 being set st
b1 . b5 = b6 & b1 . b4 in b6);
:: ZF_COLLA:prednot 2 => ZF_COLLA:pred 2
definition
let a1, a2 be set;
pred A1,A2 are_epsilon-isomorphic means
ex b1 being Relation-like Function-like set st
b1 is_epsilon-isomorphism_of a1,a2;
end;
:: ZF_COLLA:dfs 3
definiens
let a1, a2 be set;
To prove
a1,a2 are_epsilon-isomorphic
it is sufficient to prove
thus ex b1 being Relation-like Function-like set st
b1 is_epsilon-isomorphism_of a1,a2;
:: ZF_COLLA:def 3
theorem
for b1, b2 being set holds
b1,b2 are_epsilon-isomorphic
iff
ex b3 being Relation-like Function-like set st
b3 is_epsilon-isomorphism_of b1,b2;
:: ZF_COLLA:th 12
theorem
for b1 being non empty set
for b2 being Relation-like Function-like set
st proj1 b2 = b1 &
(for b3 being Element of b1 holds
b2 . b3 = b2 .: b3)
holds proj2 b2 is epsilon-transitive;
:: ZF_COLLA:th 13
theorem
for b1 being non empty set
st b1 |= the_axiom_of_extensionality
for b2, b3 being Element of b1
st for b4 being Element of b1 holds
b4 in b2
iff
b4 in b3
holds b2 = b3;
:: ZF_COLLA:th 14
theorem
for b1 being non empty set
st b1 |= the_axiom_of_extensionality
holds ex b2 being set st
b2 is epsilon-transitive & b1,b2 are_epsilon-isomorphic;