let C be Category; :: thesis: for O being non empty Subset of the carrier of C
for M being non empty set st M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } holds
( the Source of C | M is Function of M,O & the Target of C | M is Function of M,O & the Comp of C || M is PartFunc of [:M,M:],M )
let O be non empty Subset of the carrier of C; :: thesis: for M being non empty set st M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } holds
( the Source of C | M is Function of M,O & the Target of C | M is Function of M,O & the Comp of C || M is PartFunc of [:M,M:],M )
set H = { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } ;
A1: dom the Source of C = the carrier' of C by FUNCT_2:def 1;
let M be non empty set ; :: thesis: ( M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } implies ( the Source of C | M is Function of M,O & the Target of C | M is Function of M,O & the Comp of C || M is PartFunc of [:M,M:],M ) )
assume A2: M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } ; :: thesis: ( the Source of C | M is Function of M,O & the Target of C | M is Function of M,O & the Comp of C || M is PartFunc of [:M,M:],M )

A3: now :: thesis: for f being Morphism of C st f in M holds
( dom f in O & cod f in O )

let f be Morphism of C; :: thesis: ( f in M implies ( dom f in O & cod f in O ) )
assume f in M ; :: thesis: ( dom f in O & cod f in O )
then consider X being set such that
A4: f in X and
A5: X in { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } by A2, TARSKI:def 4;
ex a, b being Object of C st
( X = Hom (a,b) & a in O & b in O ) by A5;
hence ( dom f in O & cod f in O ) by A4, CAT_1:1; :: thesis: verum

end;

set d = the Source of C | M;
set c = the Target of C | M;
set p = the Comp of C || M;
A6: dom the Target of C = the carrier' of C by FUNCT_2:def 1;
A7: dom ( the Source of C | M) = (dom the Source of C) /\ M by RELAT_1:61;
A8: rng ( the Source of C | M) c= O

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ( the Source of C | M) or y in O )
assume y in rng ( the Source of C | M) ; :: thesis: y in O
then consider x being object such that
A9: x in dom ( the Source of C | M) and
A10: y = ( the Source of C | M) . x by FUNCT_1:def 3;
reconsider f = x as Morphism of C by A1, A7, A9, XBOOLE_0:def 4;
( ( the Source of C | M) . f = dom f & f in M ) by A7, A9, FUNCT_1:47, XBOOLE_0:def 4;
hence y in O by A3, A10; :: thesis: verum

end;

A11: M is non empty Subset of the carrier' of C by A2, Th15;
then A12: dom ( the Target of C | M) = M by A6, RELAT_1:62;
A13: dom ( the Target of C | M) = (dom the Target of C) /\ M by RELAT_1:61;
A14: rng ( the Target of C | M) c= O

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ( the Target of C | M) or y in O )
assume y in rng ( the Target of C | M) ; :: thesis: y in O
then consider x being object such that
A15: x in dom ( the Target of C | M) and
A16: y = ( the Target of C | M) . x by FUNCT_1:def 3;
reconsider f = x as Morphism of C by A6, A13, A15, XBOOLE_0:def 4;
( ( the Target of C | M) . f = cod f & f in M ) by A13, A15, FUNCT_1:47, XBOOLE_0:def 4;
hence y in O by A3, A16; :: thesis: verum

end;

dom ( the Source of C | M) = M by A1, A11, RELAT_1:62;
hence ( the Source of C | M is Function of M,O & the Target of C | M is Function of M,O ) by A8, A14, A12, FUNCT_2:def 1, RELSET_1:4; :: thesis: the Comp of C || M is PartFunc of [:M,M:],M
A17: dom ( the Comp of C || M) = (dom the Comp of C) /\ [:M,M:] by RELAT_1:61;
A18: dom the Comp of C c= [: the carrier' of C, the carrier' of C:] by RELAT_1:def 18;
rng ( the Comp of C || M) c= M

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng ( the Comp of C || M) or y in M )
assume y in rng ( the Comp of C || M) ; :: thesis: y in M
then consider x being object such that
A19: x in dom ( the Comp of C || M) and
A20: y = ( the Comp of C || M) . x by FUNCT_1:def 3;
A21: x in dom the Comp of C by A17, A19, XBOOLE_0:def 4;
then consider g, f being Morphism of C such that
A22: x = [g,f] by A18, DOMAIN_1:1;
A23: [g,f] in [:M,M:] by A17, A19, A22, XBOOLE_0:def 4;
then g in M by ZFMISC_1:87;
then A24: cod g in O by A3;
dom g = cod f by A21, A22, CAT_1:15;
then A25: ( dom (g (*) f) = dom f & cod (g (*) f) = cod g ) by CAT_1:17;
f in M by A23, ZFMISC_1:87;
then dom f in O by A3;
then A26: Hom ((dom (g (*) f)),(cod (g (*) f))) in { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } by A24, A25;
A27: g (*) f in Hom ((dom (g (*) f)),(cod (g (*) f))) ;
( the Comp of C || M) . x = the Comp of C . (g,f) by A19, A22, FUNCT_1:47
.= g (*) f by A21, A22, CAT_1:def 1 ;
hence y in M by A2, A20, A26, A27, TARSKI:def 4; :: thesis: verum

end;

hence the Comp of C || M is PartFunc of [:M,M:],M by A17, RELSET_1:4, XBOOLE_1:17; :: thesis: verum
thus verum ; :: thesis: verum