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Metamath Proof Explorer

Theorem funcoppc5

Description: A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 14-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 )
funcoppc5.f 
|- ( ph -> ( oppFunc ` F ) e. ( O Func P ) )
Assertion funcoppc5 
|- ( ph -> 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 funcoppc5.f 
 |-  ( ph -> ( oppFunc ` F ) e. ( O Func P ) )
6 relfunc 
 |-  Rel ( O Func P )
7 eqid 
 |-  ( oppFunc ` F ) = ( oppFunc ` F )
8 5 6 7 oppfrcl 
 |-  ( ph -> F e. ( _V X. _V ) )
9 1st2nd2 
 |-  ( F e. ( _V X. _V ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
10 8 9 syl 
 |-  ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
11 10 fveq2d 
 |-  ( ph -> ( oppFunc ` F ) = ( oppFunc ` <. ( 1st ` F ) , ( 2nd ` F ) >. ) )
12 df-ov 
 |-  ( ( 1st ` F ) oppFunc ( 2nd ` F ) ) = ( oppFunc ` <. ( 1st ` F ) , ( 2nd ` F ) >. )
13 11 12 eqtr4di 
 |-  ( ph -> ( oppFunc ` F ) = ( ( 1st ` F ) oppFunc ( 2nd ` F ) ) )
14 13 5 eqeltrrd 
 |-  ( ph -> ( ( 1st ` F ) oppFunc ( 2nd ` F ) ) e. ( O Func P ) )
15 1 2 3 4 14 funcoppc4 
 |-  ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
16 df-br 
 |-  ( ( 1st ` F ) ( C Func D ) ( 2nd ` F ) <-> <. ( 1st ` F ) , ( 2nd ` F ) >. e. ( C Func D ) )
17 15 16 sylib 
 |-  ( ph -> <. ( 1st ` F ) , ( 2nd ` F ) >. e. ( C Func D ) )
18 10 17 eqeltrd 
 |-  ( ph -> F e. ( C Func D ) )