let A, B be non empty AltGraph ; :: thesis: for F being Covariant FunctorStr of A,B st F is surjective holds
for a, b being object of B st <^a,b^> <> {} holds
for f being Morphism of a,b ex c, d being object of A ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )
let F be Covariant FunctorStr of A,B; :: thesis: ( F is surjective implies for a, b being object of B st <^a,b^> <> {} holds
for f being Morphism of a,b ex c, d being object of A ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )
given f being ManySortedFunction of the Arrows of A,the Arrows of B * the ObjectMap of F such that A1: ( f = the MorphMap of F & f is "onto" ) ; :: according to FUNCTOR0:def 33,FUNCTOR0:def 35 :: thesis: ( not F is onto or for a, b being object of B st <^a,b^> <> {} holds
for f being Morphism of a,b ex c, d being object of A ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )
assume A2: rng the ObjectMap of F = [:the carrier of B,the carrier of B:] ; :: according to FUNCTOR0:def 8,FUNCT_2:def 3 :: thesis: for a, b being object of B st <^a,b^> <> {} holds
for f being Morphism of a,b ex c, d being object of A ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )
let a, b be object of B; :: thesis: ( <^a,b^> <> {} implies for f being Morphism of a,b ex c, d being object of A ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g ) )
assume A3: <^a,b^> <> {} ; :: thesis: for f being Morphism of a,b ex c, d being object of A ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )
the ObjectMap of F is Covariant by FUNCTOR0:def 13;
then consider g being Function of the carrier of A,the carrier of B such that
A4: the ObjectMap of F = [:g,g:] by FUNCTOR0:def 2;
let f be Morphism of a,b; :: thesis: ex c, d being object of A ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )
( dom the ObjectMap of F = [:the carrier of A,the carrier of A:] & [a,b] in rng the ObjectMap of F ) by A2, FUNCT_2:def 1, ZFMISC_1:def 2;
then consider x being set such that
A5: x in [:the carrier of A,the carrier of A:] and
A6: [a,b] = the ObjectMap of F . x by FUNCT_1:def 5;
consider c, d being set such that
A7: ( c in the carrier of A & d in the carrier of A ) and
A8: [c,d] = x by A5, ZFMISC_1:def 2;
reconsider c = c, d = d as object of A by A7;
the ObjectMap of F . d,d = [(g . d),(g . d)] by A4, FUNCT_3:96;
then A9: F . d = g . d by MCART_1:7;
the ObjectMap of F . c,c = [(g . c),(g . c)] by A4, FUNCT_3:96;
then F . c = g . c by MCART_1:7;
then A10: the ObjectMap of F . c,d = [(F . c),(F . d)] by A4, A9, FUNCT_3:96;
rng (Morph-Map F,c,d) = (the Arrows of B * the ObjectMap of F) . [c,d] by A1, A5, A8, MSUALG_3:def 3
.= <^a,b^> by A5, A6, A8, FUNCT_2:21 ;
then consider g being set such that
A11: g in dom (Morph-Map F,c,d) and
A12: f = (Morph-Map F,c,d) . g by A3, FUNCT_1:def 5;
take c ; :: thesis: ex d being object of A ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )
take d ; :: thesis: ex g being Morphism of c,d st
( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )
reconsider g = g as Morphism of c,d by A11;
take g ; :: thesis: ( a = F . c & b = F . d & <^c,d^> <> {} & f = F . g )
thus ( a = F . c & b = F . d & <^c,d^> <> {} ) by A6, A8, A10, A11, ZFMISC_1:33; :: thesis: f = F . g
thus f = F . g by A3, A6, A8, A10, A11, A12, FUNCTOR0:def 16; :: thesis: verum