Metamath Proof Explorer

Theorem prf2fval

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 )
Assertion prf2fval 
|- ( ph -> ( X ( 2nd ` P ) Y ) = ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) )

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 1 2 3 4 5 prfval 
 |-  ( ph -> P = <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. )
9 2 fvexi 
 |-  B e. _V
10 9 mptex 
 |-  ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) e. _V
11 9 9 mpoex 
 |-  ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) e. _V
12 10 11 op2ndd 
 |-  ( P = <. ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. -> ( 2nd ` P ) = ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) )
13 8 12 syl 
 |-  ( ph -> ( 2nd ` P ) = ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) )
14 simprl 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> x = X )
15 simprr 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> y = Y )
16 14 15 oveq12d 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x H y ) = ( X H Y ) )
17 14 15 oveq12d 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x ( 2nd ` F ) y ) = ( X ( 2nd ` F ) Y ) )
18 17 fveq1d 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( x ( 2nd ` F ) y ) ` h ) = ( ( X ( 2nd ` F ) Y ) ` h ) )
19 14 15 oveq12d 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x ( 2nd ` G ) y ) = ( X ( 2nd ` G ) Y ) )
20 19 fveq1d 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( x ( 2nd ` G ) y ) ` h ) = ( ( X ( 2nd ` G ) Y ) ` h ) )
21 18 20 opeq12d 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. = <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. )
22 16 21 mpteq12dv 
 |-  ( ( ph /\ ( x = X /\ y = Y ) ) -> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) = ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) )
23 ovex 
 |-  ( X H Y ) e. _V
24 23 mptex 
 |-  ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) e. _V
25 24 a1i 
 |-  ( ph -> ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) e. _V )
26 13 22 6 7 25 ovmpod 
 |-  ( ph -> ( X ( 2nd ` P ) Y ) = ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) )