Article RELAT_2, MML version 4.99.1005
:: RELAT_2:prednot 1 => RELAT_2:pred 1
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_reflexive_in A2 means
for b1 being set
st b1 in a2
holds [b1,b1] in a1;
end;
:: RELAT_2:dfs 1
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_reflexive_in a2
it is sufficient to prove
thus for b1 being set
st b1 in a2
holds [b1,b1] in a1;
:: RELAT_2:def 1
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_reflexive_in b2
iff
for b3 being set
st b3 in b2
holds [b3,b3] in b1;
:: RELAT_2:prednot 2 => RELAT_2:pred 2
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_irreflexive_in A2 means
for b1 being set
st b1 in a2
holds not [b1,b1] in a1;
end;
:: RELAT_2:dfs 2
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_irreflexive_in a2
it is sufficient to prove
thus for b1 being set
st b1 in a2
holds not [b1,b1] in a1;
:: RELAT_2:def 2
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_irreflexive_in b2
iff
for b3 being set
st b3 in b2
holds not [b3,b3] in b1;
:: RELAT_2:prednot 3 => RELAT_2:pred 3
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_symmetric_in A2 means
for b1, b2 being set
st b1 in a2 & b2 in a2 & [b1,b2] in a1
holds [b2,b1] in a1;
end;
:: RELAT_2:dfs 3
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_symmetric_in a2
it is sufficient to prove
thus for b1, b2 being set
st b1 in a2 & b2 in a2 & [b1,b2] in a1
holds [b2,b1] in a1;
:: RELAT_2:def 3
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_symmetric_in b2
iff
for b3, b4 being set
st b3 in b2 & b4 in b2 & [b3,b4] in b1
holds [b4,b3] in b1;
:: RELAT_2:prednot 4 => RELAT_2:pred 4
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_antisymmetric_in A2 means
for b1, b2 being set
st b1 in a2 & b2 in a2 & [b1,b2] in a1 & [b2,b1] in a1
holds b1 = b2;
end;
:: RELAT_2:dfs 4
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_antisymmetric_in a2
it is sufficient to prove
thus for b1, b2 being set
st b1 in a2 & b2 in a2 & [b1,b2] in a1 & [b2,b1] in a1
holds b1 = b2;
:: RELAT_2:def 4
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_antisymmetric_in b2
iff
for b3, b4 being set
st b3 in b2 & b4 in b2 & [b3,b4] in b1 & [b4,b3] in b1
holds b3 = b4;
:: RELAT_2:prednot 5 => RELAT_2:pred 5
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_asymmetric_in A2 means
for b1, b2 being set
st b1 in a2 & b2 in a2 & [b1,b2] in a1
holds not [b2,b1] in a1;
end;
:: RELAT_2:dfs 5
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_asymmetric_in a2
it is sufficient to prove
thus for b1, b2 being set
st b1 in a2 & b2 in a2 & [b1,b2] in a1
holds not [b2,b1] in a1;
:: RELAT_2:def 5
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_asymmetric_in b2
iff
for b3, b4 being set
st b3 in b2 & b4 in b2 & [b3,b4] in b1
holds not [b4,b3] in b1;
:: RELAT_2:prednot 6 => RELAT_2:pred 6
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_connected_in A2 means
for b1, b2 being set
st b1 in a2 & b2 in a2 & b1 <> b2 & not [b1,b2] in a1
holds [b2,b1] in a1;
end;
:: RELAT_2:dfs 6
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_connected_in a2
it is sufficient to prove
thus for b1, b2 being set
st b1 in a2 & b2 in a2 & b1 <> b2 & not [b1,b2] in a1
holds [b2,b1] in a1;
:: RELAT_2:def 6
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_connected_in b2
iff
for b3, b4 being set
st b3 in b2 & b4 in b2 & b3 <> b4 & not [b3,b4] in b1
holds [b4,b3] in b1;
:: RELAT_2:prednot 7 => RELAT_2:pred 7
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_strongly_connected_in A2 means
for b1, b2 being set
st b1 in a2 & b2 in a2 & not [b1,b2] in a1
holds [b2,b1] in a1;
end;
:: RELAT_2:dfs 7
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_strongly_connected_in a2
it is sufficient to prove
thus for b1, b2 being set
st b1 in a2 & b2 in a2 & not [b1,b2] in a1
holds [b2,b1] in a1;
:: RELAT_2:def 7
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_strongly_connected_in b2
iff
for b3, b4 being set
st b3 in b2 & b4 in b2 & not [b3,b4] in b1
holds [b4,b3] in b1;
:: RELAT_2:prednot 8 => RELAT_2:pred 8
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 is_transitive_in A2 means
for b1, b2, b3 being set
st b1 in a2 & b2 in a2 & b3 in a2 & [b1,b2] in a1 & [b2,b3] in a1
holds [b1,b3] in a1;
end;
:: RELAT_2:dfs 8
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 is_transitive_in a2
it is sufficient to prove
thus for b1, b2, b3 being set
st b1 in a2 & b2 in a2 & b3 in a2 & [b1,b2] in a1 & [b2,b3] in a1
holds [b1,b3] in a1;
:: RELAT_2:def 8
theorem
for b1 being Relation-like set
for b2 being set holds
b1 is_transitive_in b2
iff
for b3, b4, b5 being set
st b3 in b2 & b4 in b2 & b5 in b2 & [b3,b4] in b1 & [b4,b5] in b1
holds [b3,b5] in b1;
:: RELAT_2:attrnot 1 => RELAT_2:attr 1
definition
let a1 be Relation-like set;
attr a1 is reflexive means
a1 is_reflexive_in field a1;
end;
:: RELAT_2:dfs 9
definiens
let a1 be Relation-like set;
To prove
a1 is reflexive
it is sufficient to prove
thus a1 is_reflexive_in field a1;
:: RELAT_2:def 9
theorem
for b1 being Relation-like set holds
b1 is reflexive
iff
b1 is_reflexive_in field b1;
:: RELAT_2:attrnot 2 => RELAT_2:attr 2
definition
let a1 be Relation-like set;
attr a1 is irreflexive means
a1 is_irreflexive_in field a1;
end;
:: RELAT_2:dfs 10
definiens
let a1 be Relation-like set;
To prove
a1 is irreflexive
it is sufficient to prove
thus a1 is_irreflexive_in field a1;
:: RELAT_2:def 10
theorem
for b1 being Relation-like set holds
b1 is irreflexive
iff
b1 is_irreflexive_in field b1;
:: RELAT_2:attrnot 3 => RELAT_2:attr 3
definition
let a1 be Relation-like set;
attr a1 is symmetric means
a1 is_symmetric_in field a1;
end;
:: RELAT_2:dfs 11
definiens
let a1 be Relation-like set;
To prove
a1 is symmetric
it is sufficient to prove
thus a1 is_symmetric_in field a1;
:: RELAT_2:def 11
theorem
for b1 being Relation-like set holds
b1 is symmetric
iff
b1 is_symmetric_in field b1;
:: RELAT_2:attrnot 4 => RELAT_2:attr 4
definition
let a1 be Relation-like set;
attr a1 is antisymmetric means
a1 is_antisymmetric_in field a1;
end;
:: RELAT_2:dfs 12
definiens
let a1 be Relation-like set;
To prove
a1 is antisymmetric
it is sufficient to prove
thus a1 is_antisymmetric_in field a1;
:: RELAT_2:def 12
theorem
for b1 being Relation-like set holds
b1 is antisymmetric
iff
b1 is_antisymmetric_in field b1;
:: RELAT_2:attrnot 5 => RELAT_2:attr 5
definition
let a1 be Relation-like set;
attr a1 is asymmetric means
a1 is_asymmetric_in field a1;
end;
:: RELAT_2:dfs 13
definiens
let a1 be Relation-like set;
To prove
a1 is asymmetric
it is sufficient to prove
thus a1 is_asymmetric_in field a1;
:: RELAT_2:def 13
theorem
for b1 being Relation-like set holds
b1 is asymmetric
iff
b1 is_asymmetric_in field b1;
:: RELAT_2:attrnot 6 => RELAT_2:attr 6
definition
let a1 be Relation-like set;
attr a1 is connected means
a1 is_connected_in field a1;
end;
:: RELAT_2:dfs 14
definiens
let a1 be Relation-like set;
To prove
a1 is connected
it is sufficient to prove
thus a1 is_connected_in field a1;
:: RELAT_2:def 14
theorem
for b1 being Relation-like set holds
b1 is connected
iff
b1 is_connected_in field b1;
:: RELAT_2:attrnot 7 => RELAT_2:attr 7
definition
let a1 be Relation-like set;
attr a1 is strongly_connected means
a1 is_strongly_connected_in field a1;
end;
:: RELAT_2:dfs 15
definiens
let a1 be Relation-like set;
To prove
a1 is strongly_connected
it is sufficient to prove
thus a1 is_strongly_connected_in field a1;
:: RELAT_2:def 15
theorem
for b1 being Relation-like set holds
b1 is strongly_connected
iff
b1 is_strongly_connected_in field b1;
:: RELAT_2:attrnot 8 => RELAT_2:attr 8
definition
let a1 be Relation-like set;
attr a1 is transitive means
a1 is_transitive_in field a1;
end;
:: RELAT_2:dfs 16
definiens
let a1 be Relation-like set;
To prove
a1 is transitive
it is sufficient to prove
thus a1 is_transitive_in field a1;
:: RELAT_2:def 16
theorem
for b1 being Relation-like set holds
b1 is transitive
iff
b1 is_transitive_in field b1;
:: RELAT_2:th 17
theorem
for b1 being Relation-like set holds
b1 is reflexive
iff
id field b1 c= b1;
:: RELAT_2:th 18
theorem
for b1 being Relation-like set holds
b1 is irreflexive
iff
id field b1 misses b1;
:: RELAT_2:th 19
theorem
for b1 being set
for b2 being Relation-like set holds
b2 is_antisymmetric_in b1
iff
b2 \ id b1 is_asymmetric_in b1;
:: RELAT_2:th 20
theorem
for b1 being set
for b2 being Relation-like set
st b2 is_asymmetric_in b1
holds b2 \/ id b1 is_antisymmetric_in b1;
:: RELAT_2:th 22
theorem
for b1 being Relation-like set
st b1 is symmetric & b1 is transitive
holds b1 is reflexive;
:: RELAT_2:th 23
theorem
for b1 being set holds
id b1 is symmetric & id b1 is transitive;
:: RELAT_2:th 24
theorem
for b1 being set holds
id b1 is antisymmetric & id b1 is reflexive;
:: RELAT_2:th 25
theorem
for b1 being Relation-like set
st b1 is irreflexive & b1 is transitive
holds b1 is asymmetric;
:: RELAT_2:th 26
theorem
for b1 being Relation-like set
st b1 is asymmetric
holds b1 is irreflexive & b1 is antisymmetric;
:: RELAT_2:th 27
theorem
for b1 being Relation-like set
st b1 is reflexive
holds b1 ~ is reflexive;
:: RELAT_2:th 28
theorem
for b1 being Relation-like set
st b1 is irreflexive
holds b1 ~ is irreflexive;
:: RELAT_2:th 29
theorem
for b1 being Relation-like set
st b1 is reflexive
holds proj1 b1 = proj1 (b1 ~) & proj2 b1 = proj2 (b1 ~);
:: RELAT_2:th 30
theorem
for b1 being Relation-like set holds
b1 is symmetric
iff
b1 = b1 ~;
:: RELAT_2:th 31
theorem
for b1, b2 being Relation-like set
st b1 is reflexive & b2 is reflexive
holds b1 \/ b2 is reflexive & b1 /\ b2 is reflexive;
:: RELAT_2:th 32
theorem
for b1, b2 being Relation-like set
st b1 is irreflexive & b2 is irreflexive
holds b1 \/ b2 is irreflexive & b1 /\ b2 is irreflexive;
:: RELAT_2:th 33
theorem
for b1, b2 being Relation-like set
st b1 is irreflexive
holds b1 \ b2 is irreflexive;
:: RELAT_2:th 34
theorem
for b1 being Relation-like set
st b1 is symmetric
holds b1 ~ is symmetric;
:: RELAT_2:th 35
theorem
for b1, b2 being Relation-like set
st b1 is symmetric & b2 is symmetric
holds b1 \/ b2 is symmetric & b1 /\ b2 is symmetric & b1 \ b2 is symmetric;
:: RELAT_2:th 36
theorem
for b1 being Relation-like set
st b1 is asymmetric
holds b1 ~ is asymmetric;
:: RELAT_2:th 37
theorem
for b1, b2 being Relation-like set
st b1 is asymmetric & b2 is asymmetric
holds b1 /\ b2 is asymmetric;
:: RELAT_2:th 38
theorem
for b1, b2 being Relation-like set
st b1 is asymmetric
holds b1 \ b2 is asymmetric;
:: RELAT_2:th 39
theorem
for b1 being Relation-like set holds
b1 is antisymmetric
iff
b1 /\ (b1 ~) c= id proj1 b1;
:: RELAT_2:th 40
theorem
for b1 being Relation-like set
st b1 is antisymmetric
holds b1 ~ is antisymmetric;
:: RELAT_2:th 41
theorem
for b1, b2 being Relation-like set
st b1 is antisymmetric
holds b1 /\ b2 is antisymmetric & b1 \ b2 is antisymmetric;
:: RELAT_2:th 42
theorem
for b1 being Relation-like set
st b1 is transitive
holds b1 ~ is transitive;
:: RELAT_2:th 43
theorem
for b1, b2 being Relation-like set
st b1 is transitive & b2 is transitive
holds b1 /\ b2 is transitive;
:: RELAT_2:th 44
theorem
for b1 being Relation-like set holds
b1 is transitive
iff
b1 * b1 c= b1;
:: RELAT_2:th 45
theorem
for b1 being Relation-like set holds
b1 is connected
iff
[:field b1,field b1:] \ id field b1 c= b1 \/ (b1 ~);
:: RELAT_2:th 46
theorem
for b1 being Relation-like set
st b1 is strongly_connected
holds b1 is connected & b1 is reflexive;
:: RELAT_2:th 47
theorem
for b1 being Relation-like set holds
b1 is strongly_connected
iff
[:field b1,field b1:] = b1 \/ (b1 ~);
:: RELAT_2:th 48
theorem
for b1 being Relation-like set holds
b1 is transitive
iff
for b2, b3, b4 being set
st [b2,b3] in b1 & [b3,b4] in b1
holds [b2,b4] in b1;