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 doms F = I --> (cod f) holds

( doms (F * f) = I --> (dom f) & cods (F * f) = cods F )

let C be Category; :: thesis: for f being Morphism of C

for F being Function of I, the carrier' of C st doms F = I --> (cod f) holds

( doms (F * f) = I --> (dom f) & cods (F * f) = cods F )

let f be Morphism of C; :: thesis: for F being Function of I, the carrier' of C st doms F = I --> (cod f) holds

( doms (F * f) = I --> (dom f) & cods (F * f) = cods F )

let F be Function of I, the carrier' of C; :: thesis: ( doms F = I --> (cod f) implies ( doms (F * f) = I --> (dom f) & cods (F * f) = cods F ) )

assume A1: doms F = I --> (cod f) ; :: thesis: ( doms (F * f) = I --> (dom f) & cods (F * f) = cods F )

for f being Morphism of C

for F being Function of I, the carrier' of C st doms F = I --> (cod f) holds

( doms (F * f) = I --> (dom f) & cods (F * f) = cods F )

let C be Category; :: thesis: for f being Morphism of C

for F being Function of I, the carrier' of C st doms F = I --> (cod f) holds

( doms (F * f) = I --> (dom f) & cods (F * f) = cods F )

let f be Morphism of C; :: thesis: for F being Function of I, the carrier' of C st doms F = I --> (cod f) holds

( doms (F * f) = I --> (dom f) & cods (F * f) = cods F )

let F be Function of I, the carrier' of C; :: thesis: ( doms F = I --> (cod f) implies ( doms (F * f) = I --> (dom f) & cods (F * f) = cods F ) )

assume A1: doms F = I --> (cod f) ; :: thesis: ( doms (F * f) = I --> (dom f) & cods (F * f) = cods F )

now :: thesis: for x being set st x in I holds

(doms (F * f)) /. x = (I --> (dom f)) /. x

hence
doms (F * f) = I --> (dom f)
by Th1; :: thesis: cods (F * f) = cods F(doms (F * f)) /. x = (I --> (dom f)) /. x

let x be set ; :: thesis: ( x in I implies (doms (F * f)) /. x = (I --> (dom f)) /. x )

assume A2: x in I ; :: thesis: (doms (F * f)) /. x = (I --> (dom f)) /. x

then A3: dom (F /. x) = (I --> (cod f)) /. x by A1, Def1

.= cod f by A2, Th2 ;

thus (doms (F * f)) /. x = dom ((F * f) /. x) by A2, Def1

.= dom ((F /. x) (*) f) by A2, Def5

.= dom f by A3, CAT_1:17

.= (I --> (dom f)) /. x by A2, Th2 ; :: thesis: verum

end;assume A2: x in I ; :: thesis: (doms (F * f)) /. x = (I --> (dom f)) /. x

then A3: dom (F /. x) = (I --> (cod f)) /. x by A1, Def1

.= cod f by A2, Th2 ;

thus (doms (F * f)) /. x = dom ((F * f) /. x) by A2, Def1

.= dom ((F /. x) (*) f) by A2, Def5

.= dom f by A3, CAT_1:17

.= (I --> (dom f)) /. x by A2, Th2 ; :: thesis: verum

now :: thesis: for x being set st x in I holds

(cods F) /. x = (cods (F * f)) /. x

hence
cods (F * f) = cods F
by Th1; :: thesis: verum(cods F) /. x = (cods (F * f)) /. x

let x be set ; :: thesis: ( x in I implies (cods F) /. x = (cods (F * f)) /. x )

assume A4: x in I ; :: thesis: (cods F) /. x = (cods (F * f)) /. x

then A5: dom (F /. x) = (I --> (cod f)) /. x by A1, Def1

.= cod f by A4, Th2 ;

thus (cods F) /. x = cod (F /. x) by A4, Def2

.= cod ((F /. x) (*) f) by A5, CAT_1:17

.= cod ((F * f) /. x) by A4, Def5

.= (cods (F * f)) /. x by A4, Def2 ; :: thesis: verum

end;assume A4: x in I ; :: thesis: (cods F) /. x = (cods (F * f)) /. x

then A5: dom (F /. x) = (I --> (cod f)) /. x by A1, Def1

.= cod f by A4, Th2 ;

thus (cods F) /. x = cod (F /. x) by A4, Def2

.= cod ((F /. x) (*) f) by A5, CAT_1:17

.= cod ((F * f) /. x) by A4, Def5

.= (cods (F * f)) /. x by A4, Def2 ; :: thesis: verum