let A, B be non empty transitive with_units AltCatStr ; :: thesis: for F being reflexive feasible id-preserving FunctorStr over A,B st F is bijective & F is coreflexive holds
F " is id-preserving
let F be reflexive feasible id-preserving FunctorStr over A,B; :: thesis: ( F is bijective & F is coreflexive implies F " is id-preserving )
assume A1: ( F is bijective & F is coreflexive ) ; :: thesis: F " is id-preserving
set G = F " ;
reconsider H = F " as reflexive feasible FunctorStr over B,A by A1, Th35, Th36;
A2: the ObjectMap of (F ") = the ObjectMap of F " by A1, Def38;
consider f being ManySortedFunction of the Arrows of A, the Arrows of B * the ObjectMap of F such that
A3: f = the MorphMap of F and
A4: the MorphMap of (F ") = (f "") * ( the ObjectMap of F ") by A1, Def38;
let o be Object of B; :: according to FUNCTOR0:def 20 :: thesis: (Morph-Map ((F "),o,o)) . (idm o) = idm ((F ") . o)
A5: F is injective by A1;
then F is one-to-one ;
then A6: the ObjectMap of F is one-to-one ;
F is faithful by A5;
then A7: the MorphMap of F is "1-1" ;
F is surjective by A1;
then F is full ;
then A8: ex f being ManySortedFunction of the Arrows of A, the Arrows of B * the ObjectMap of F st
( f = the MorphMap of F & f is "onto" ) ;
A9: [((F ") . o),((F ") . o)] in [: the carrier of A, the carrier of A:] by ZFMISC_1:87;
A10: [o,o] in [: the carrier of B, the carrier of B:] by ZFMISC_1:87;
then A11: [o,o] in dom the ObjectMap of (F ") by FUNCT_2:def 1;
A12: the ObjectMap of (F * H) . (o,o) = ( the ObjectMap of F * the ObjectMap of H) . [o,o] by Def36
.= ( the ObjectMap of F * ( the ObjectMap of F ")) . [o,o] by A1, Def38
.= (id (rng the ObjectMap of F)) . [o,o] by A6, FUNCT_1:39
.= (id (dom the ObjectMap of (F "))) . [o,o] by A2, A6, FUNCT_1:33
.= (id [: the carrier of B, the carrier of B:]) . [o,o] by FUNCT_2:def 1
.= [o,o] by A10, FUNCT_1:18 ;
A13: F . ((F ") . o) = (F * H) . o by Th33
.= o by A12 ;
dom the MorphMap of F = [: the carrier of A, the carrier of A:] by PARTFUN1:def 2;
then [((F ") . o),((F ") . o)] in dom the MorphMap of F by ZFMISC_1:87;
then A14: Morph-Map (F,((F ") . o),((F ") . o)) is one-to-one by A7;
[((F ") . o),((F ") . o)] in dom the ObjectMap of F by A9, FUNCT_2:def 1;
then ( the Arrows of B * the ObjectMap of F) . [((F ") . o),((F ") . o)] = the Arrows of B . ( the ObjectMap of F . (((F ") . o),((F ") . o))) by FUNCT_1:13
.= the Arrows of B . ((F . ((F ") . o)),(F . ((F ") . o))) by Def10 ;
then A15: ( the Arrows of B * the ObjectMap of F) . [((F ") . o),((F ") . o)] <> {} by ALTCAT_2:def 6;
Morph-Map (F,((F ") . o),((F ") . o)) is Function of ( the Arrows of A . [((F ") . o),((F ") . o)]),(( the Arrows of B * the ObjectMap of F) . [((F ") . o),((F ") . o)]) by A3, A9, PBOOLE:def 15;
then A16: dom (Morph-Map (F,((F ") . o),((F ") . o))) = the Arrows of A . (((F ") . o),((F ") . o)) by A15, FUNCT_2:def 1
.= <^((F ") . o),((F ") . o)^> by ALTCAT_1:def 1 ;
( the Arrows of A * the ObjectMap of (F ")) . [o,o] = the Arrows of A . ( the ObjectMap of H . (o,o)) by A11, FUNCT_1:13
.= the Arrows of A . (((F ") . o),((F ") . o)) by Def10 ;
then A17: ( the Arrows of A * the ObjectMap of (F ")) . [o,o] <> {} by ALTCAT_2:def 6;
the MorphMap of (F ") is ManySortedFunction of the Arrows of B, the Arrows of A * the ObjectMap of (F ") by Def4;
then Morph-Map ((F "),o,o) is Function of ( the Arrows of B . [o,o]),(( the Arrows of A * the ObjectMap of (F ")) . [o,o]) by A10, PBOOLE:def 15;
then A18: dom (Morph-Map ((F "),o,o)) = the Arrows of B . (o,o) by A17, FUNCT_2:def 1
.= <^o,o^> by ALTCAT_1:def 1 ;
then A19: idm o in dom (Morph-Map ((F "),o,o)) by ALTCAT_1:19;
A20: Morph-Map ((F "),o,o) = (f "") . ( the ObjectMap of (F ") . (o,o)) by A2, A4, A11, FUNCT_1:13
.= (f "") . [(H . o),(H . o)] by Def10
.= (Morph-Map (F,((F ") . o),((F ") . o))) " by A3, A7, A8, A9, MSUALG_3:def 4 ;
(Morph-Map ((F "),o,o)) . (idm o) in rng (Morph-Map ((F "),o,o)) by A19, FUNCT_1:def 3;
then A21: (Morph-Map ((F "),o,o)) . (idm o) in dom (Morph-Map (F,((F ") . o),((F ") . o))) by A14, A20, FUNCT_1:33;
(Morph-Map (F,((F ") . o),((F ") . o))) . ((Morph-Map ((F "),o,o)) . (idm o)) = ((Morph-Map (F,((F ") . o),((F ") . o))) * (Morph-Map ((F "),o,o))) . (idm o) by A18, ALTCAT_1:19, FUNCT_1:13
.= (id (rng (Morph-Map (F,((F ") . o),((F ") . o))))) . (idm o) by A14, A20, FUNCT_1:39
.= (id (dom (Morph-Map ((F "),o,o)))) . (idm o) by A14, A20, FUNCT_1:33
.= idm (F . ((F ") . o)) by A13, A18
.= (Morph-Map (F,((F ") . o),((F ") . o))) . (idm ((F ") . o)) by Def20 ;
hence (Morph-Map ((F "),o,o)) . (idm o) = idm ((F ") . o) by A14, A16, A21; :: thesis: verum