let I be set ; :: thesis: for C being Category
for f being Morphism of C
for F being Function of I,the carrier' of C st cods F = I --> (dom f) holds
( doms (f * F) = doms F & cods (f * F) = I --> (cod f) )
let C be Category; :: thesis: for f being Morphism of C
for F being Function of I,the carrier' of C st cods F = I --> (dom f) holds
( doms (f * F) = doms F & cods (f * F) = I --> (cod f) )
let f be Morphism of C; :: thesis: for F being Function of I,the carrier' of C st cods F = I --> (dom f) holds
( doms (f * F) = doms F & cods (f * F) = I --> (cod f) )
let F be Function of I,the carrier' of C; :: thesis: ( cods F = I --> (dom f) implies ( doms (f * F) = doms F & cods (f * F) = I --> (cod f) ) )
assume A1: cods F = I --> (dom f) ; :: thesis: ( doms (f * F) = doms F & cods (f * F) = I --> (cod f) )

now

let x be set ; :: thesis: ( x in I implies (doms F) /. x = (doms (f * F)) /. x )
assume A2: x in I ; :: thesis: (doms F) /. x = (doms (f * F)) /. x
then A3: cod (F /. x) = (I --> (dom f)) /. x by A1, Def4
.= dom f by A2, Th2 ;
thus (doms F) /. x = dom (F /. x) by A2, Def3
.= dom (f * (F /. x)) by A3, CAT_1:42
.= dom ((f * F) /. x) by A2, Def8
.= (doms (f * F)) /. x by A2, Def3 ; :: thesis: verum

end;

hence doms (f * F) = doms F by Th1; :: thesis: cods (f * F) = I --> (cod f)

now

let x be set ; :: thesis: ( x in I implies (cods (f * F)) /. x = (I --> (cod f)) /. x )
assume A4: x in I ; :: thesis: (cods (f * F)) /. x = (I --> (cod f)) /. x
then A5: cod (F /. x) = (I --> (dom f)) /. x by A1, Def4
.= dom f by A4, Th2 ;
thus (cods (f * F)) /. x = cod ((f * F) /. x) by A4, Def4
.= cod (f * (F /. x)) by A4, Def8
.= cod f by A5, CAT_1:42
.= (I --> (cod f)) /. x by A4, Th2 ; :: thesis: verum

end;

hence cods (f * F) = I --> (cod f) by Th1; :: thesis: verum