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Metamath Proof Explorer

Theorem 2oppffunc

Description: The opposite functor of an opposite functor is a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025) The functor in opposite categories does not have to be an opposite functor. (Revised by Zhi Wang, 17-Nov-2025)

Ref Expression
Hypotheses funcoppc2.o 
|- O = ( oppCat ` C )
funcoppc2.p 
|- P = ( oppCat ` D )
funcoppc2.c 
|- ( ph -> C e. V )
funcoppc2.d 
|- ( ph -> D e. W )
2oppffunc.f 
|- ( ph -> F e. ( O Func P ) )
Assertion 2oppffunc 
|- ( ph -> ( oppFunc ` F ) e. ( C Func D ) )

Proof

Step Hyp Ref Expression
1 funcoppc2.o 
 |-  O = ( oppCat ` C )
2 funcoppc2.p 
 |-  P = ( oppCat ` D )
3 funcoppc2.c 
 |-  ( ph -> C e. V )
4 funcoppc2.d 
 |-  ( ph -> D e. W )
5 2oppffunc.f 
 |-  ( ph -> F e. ( O Func P ) )
6 oppfval2 
 |-  ( F e. ( O Func P ) -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. )
7 5 6 syl 
 |-  ( ph -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. )
8 5 func1st2nd 
 |-  ( ph -> ( 1st ` F ) ( O Func P ) ( 2nd ` F ) )
9 1 2 3 4 8 funcoppc2 
 |-  ( ph -> ( 1st ` F ) ( C Func D ) tpos ( 2nd ` F ) )
10 df-br 
 |-  ( ( 1st ` F ) ( C Func D ) tpos ( 2nd ` F ) <-> <. ( 1st ` F ) , tpos ( 2nd ` F ) >. e. ( C Func D ) )
11 9 10 sylib 
 |-  ( ph -> <. ( 1st ` F ) , tpos ( 2nd ` F ) >. e. ( C Func D ) )
12 7 11 eqeltrd 
 |-  ( ph -> ( oppFunc ` F ) e. ( C Func D ) )