let C1, C2 be non empty AltCatStr ; :: thesis: for F being Contravariant FunctorStr over C1,C2 holds

( F is full iff for o1, o2 being Object of C1 holds Morph-Map (F,o2,o1) is onto )

let F be Contravariant FunctorStr over C1,C2; :: thesis: ( F is full iff for o1, o2 being Object of C1 holds Morph-Map (F,o2,o1) is onto )

set I = [: the carrier of C1, the carrier of C1:];

ex I29 being non empty set ex B9 being ManySortedSet of I29 ex f9 being Function of [: the carrier of C1, the carrier of C1:],I29 st

( the ObjectMap of F = f9 & the Arrows of C2 = B9 & the MorphMap of F is ManySortedFunction of the Arrows of C1,B9 * f9 ) by FUNCTOR0:def 3;

then reconsider f = the MorphMap of F as ManySortedFunction of the Arrows of C1, the Arrows of C2 * the ObjectMap of F ;

take f ; :: according to FUNCTOR0:def 32 :: thesis: ( f = the MorphMap of F & f is "onto" )

thus f = the MorphMap of F ; :: thesis: f is "onto"

let i be set ; :: according to MSUALG_3:def 3 :: thesis: ( not i in [: the carrier of C1, the carrier of C1:] or proj2 (f . i) = ( the ObjectMap of F * the Arrows of C2) . i )

assume i in [: the carrier of C1, the carrier of C1:] ; :: thesis: proj2 (f . i) = ( the ObjectMap of F * the Arrows of C2) . i

then consider o2, o1 being object such that

A7: ( o2 in the carrier of C1 & o1 in the carrier of C1 ) and

A8: i = [o2,o1] by ZFMISC_1:84;

reconsider o1 = o1, o2 = o2 as Object of C1 by A7;

[o2,o1] in [: the carrier of C1, the carrier of C1:] by ZFMISC_1:87;

then A9: [o2,o1] in dom the ObjectMap of F by FUNCT_2:def 1;

Morph-Map (F,o2,o1) is onto by A6;

then rng (Morph-Map (F,o2,o1)) = the Arrows of C2 . ((F . o1),(F . o2))

.= the Arrows of C2 . ( the ObjectMap of F . (o2,o1)) by FUNCTOR0:23

.= ( the Arrows of C2 * the ObjectMap of F) . [o2,o1] by A9, FUNCT_1:13 ;

hence proj2 (f . i) = ( the ObjectMap of F * the Arrows of C2) . i by A8; :: thesis: verum

( F is full iff for o1, o2 being Object of C1 holds Morph-Map (F,o2,o1) is onto )

let F be Contravariant FunctorStr over C1,C2; :: thesis: ( F is full iff for o1, o2 being Object of C1 holds Morph-Map (F,o2,o1) is onto )

set I = [: the carrier of C1, the carrier of C1:];

hereby :: thesis: ( ( for o1, o2 being Object of C1 holds Morph-Map (F,o2,o1) is onto ) implies F is full )

assume A6:
for o1, o2 being Object of C1 holds Morph-Map (F,o2,o1) is onto
; :: thesis: F is full assume A1:
F is full
; :: thesis: for o1, o2 being Object of C1 holds Morph-Map (F,o2,o1) is onto

let o1, o2 be Object of C1; :: thesis: Morph-Map (F,o2,o1) is onto

thus Morph-Map (F,o2,o1) is onto :: thesis: verum

end;let o1, o2 be Object of C1; :: thesis: Morph-Map (F,o2,o1) is onto

thus Morph-Map (F,o2,o1) is onto :: thesis: verum

proof

A2:
[o2,o1] in [: the carrier of C1, the carrier of C1:]
by ZFMISC_1:87;

then A3: [o2,o1] in dom the ObjectMap of F by FUNCT_2:def 1;

consider f being ManySortedFunction of the Arrows of C1, the Arrows of C2 * the ObjectMap of F such that

A4: f = the MorphMap of F and

A5: f is "onto" by A1;

rng (f . [o2,o1]) = ( the Arrows of C2 * the ObjectMap of F) . [o2,o1] by A5, A2;

hence rng (Morph-Map (F,o2,o1)) = the Arrows of C2 . ( the ObjectMap of F . (o2,o1)) by A4, A3, FUNCT_1:13

.= <^(F . o1),(F . o2)^> by FUNCTOR0:23 ;

:: according to FUNCT_2:def 3 :: thesis: verum

end;then A3: [o2,o1] in dom the ObjectMap of F by FUNCT_2:def 1;

consider f being ManySortedFunction of the Arrows of C1, the Arrows of C2 * the ObjectMap of F such that

A4: f = the MorphMap of F and

A5: f is "onto" by A1;

rng (f . [o2,o1]) = ( the Arrows of C2 * the ObjectMap of F) . [o2,o1] by A5, A2;

hence rng (Morph-Map (F,o2,o1)) = the Arrows of C2 . ( the ObjectMap of F . (o2,o1)) by A4, A3, FUNCT_1:13

.= <^(F . o1),(F . o2)^> by FUNCTOR0:23 ;

:: according to FUNCT_2:def 3 :: thesis: verum

ex I29 being non empty set ex B9 being ManySortedSet of I29 ex f9 being Function of [: the carrier of C1, the carrier of C1:],I29 st

( the ObjectMap of F = f9 & the Arrows of C2 = B9 & the MorphMap of F is ManySortedFunction of the Arrows of C1,B9 * f9 ) by FUNCTOR0:def 3;

then reconsider f = the MorphMap of F as ManySortedFunction of the Arrows of C1, the Arrows of C2 * the ObjectMap of F ;

take f ; :: according to FUNCTOR0:def 32 :: thesis: ( f = the MorphMap of F & f is "onto" )

thus f = the MorphMap of F ; :: thesis: f is "onto"

let i be set ; :: according to MSUALG_3:def 3 :: thesis: ( not i in [: the carrier of C1, the carrier of C1:] or proj2 (f . i) = ( the ObjectMap of F * the Arrows of C2) . i )

assume i in [: the carrier of C1, the carrier of C1:] ; :: thesis: proj2 (f . i) = ( the ObjectMap of F * the Arrows of C2) . i

then consider o2, o1 being object such that

A7: ( o2 in the carrier of C1 & o1 in the carrier of C1 ) and

A8: i = [o2,o1] by ZFMISC_1:84;

reconsider o1 = o1, o2 = o2 as Object of C1 by A7;

[o2,o1] in [: the carrier of C1, the carrier of C1:] by ZFMISC_1:87;

then A9: [o2,o1] in dom the ObjectMap of F by FUNCT_2:def 1;

Morph-Map (F,o2,o1) is onto by A6;

then rng (Morph-Map (F,o2,o1)) = the Arrows of C2 . ((F . o1),(F . o2))

.= the Arrows of C2 . ( the ObjectMap of F . (o2,o1)) by FUNCTOR0:23

.= ( the Arrows of C2 * the ObjectMap of F) . [o2,o1] by A9, FUNCT_1:13 ;

hence proj2 (f . i) = ( the ObjectMap of F * the Arrows of C2) . i by A8; :: thesis: verum