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Metamath Proof Explorer

Theorem prf2

Description: Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017)

Ref Expression
Hypotheses prfval.k 
|- P = ( F pairF G )
prfval.b 
|- B = ( Base ` C )
prfval.h 
|- H = ( Hom ` C )
prfval.c 
|- ( ph -> F e. ( C Func D ) )
prfval.d 
|- ( ph -> G e. ( C Func E ) )
prf1.x 
|- ( ph -> X e. B )
prf2.y 
|- ( ph -> Y e. B )
prf2.k 
|- ( ph -> K e. ( X H Y ) )
Assertion prf2 
|- ( ph -> ( ( X ( 2nd ` P ) Y ) ` K ) = <. ( ( X ( 2nd ` F ) Y ) ` K ) , ( ( X ( 2nd ` G ) Y ) ` K ) >. )

Proof

Step Hyp Ref Expression
1 prfval.k 
 |-  P = ( F pairF G )
2 prfval.b 
 |-  B = ( Base ` C )
3 prfval.h 
 |-  H = ( Hom ` C )
4 prfval.c 
 |-  ( ph -> F e. ( C Func D ) )
5 prfval.d 
 |-  ( ph -> G e. ( C Func E ) )
6 prf1.x 
 |-  ( ph -> X e. B )
7 prf2.y 
 |-  ( ph -> Y e. B )
8 prf2.k 
 |-  ( ph -> K e. ( X H Y ) )
9 1 2 3 4 5 6 7 prf2fval 
 |-  ( ph -> ( X ( 2nd ` P ) Y ) = ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) )
10 simpr 
 |-  ( ( ph /\ h = K ) -> h = K )
11 10 fveq2d 
 |-  ( ( ph /\ h = K ) -> ( ( X ( 2nd ` F ) Y ) ` h ) = ( ( X ( 2nd ` F ) Y ) ` K ) )
12 10 fveq2d 
 |-  ( ( ph /\ h = K ) -> ( ( X ( 2nd ` G ) Y ) ` h ) = ( ( X ( 2nd ` G ) Y ) ` K ) )
13 11 12 opeq12d 
 |-  ( ( ph /\ h = K ) -> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. = <. ( ( X ( 2nd ` F ) Y ) ` K ) , ( ( X ( 2nd ` G ) Y ) ` K ) >. )
14 opex 
 |-  <. ( ( X ( 2nd ` F ) Y ) ` K ) , ( ( X ( 2nd ` G ) Y ) ` K ) >. e. _V
15 14 a1i 
 |-  ( ph -> <. ( ( X ( 2nd ` F ) Y ) ` K ) , ( ( X ( 2nd ` G ) Y ) ` K ) >. e. _V )
16 9 13 8 15 fvmptd 
 |-  ( ph -> ( ( X ( 2nd ` P ) Y ) ` K ) = <. ( ( X ( 2nd ` F ) Y ) ` K ) , ( ( X ( 2nd ` G ) Y ) ` K ) >. )