Theorem prf2fval 16841

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

Hypotheses

Ref	Expression
prfval.k	|- P = ( F ⟨,⟩F 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

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

Distinct variable groups:   B, h   h, F   ph, h   h, G   h, X   h, Y   h, H

Allowed substitution hints:   C( h)   D( h)   P( h)   E( h)

Proof of Theorem prf2fval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prfval.k	. . . 4 |- P = ( F ⟨,⟩F G )
2		prfval.b	. . . 4 |- B = ( Base ` C )
3		prfval.h	. . . 4 |- H = ( Hom ` C )
4		prfval.c	. . . 4 |- ( ph -> F e. ( C Func D ) )
5		prfval.d	. . . 4 |- ( ph -> G e. ( C Func E ) )
6	1, 2, 3, 4, 5	prfval 16839	. . 3 |- ( 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 ) >. ) ) >. )
7		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
8	2, 7	eqeltri 2697	. . . . 5 |- B e. _V
9	8	mptex 6486	. . . 4 |- ( x e. B |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) e. _V
10	8, 8	mpt2ex 7247	. . . 4 |- ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) e. _V
11	9, 10	op2ndd 7179	. . 3 |- ( 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 ) >. ) ) )
12	6, 11	syl 17	. 2 |- ( ph -> ( 2nd ` P ) = ( x e. B , y e. B |-> ( h e. ( x H y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) )
13		simprl 794	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> x = X )
14		simprr 796	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> y = Y )
15	13, 14	oveq12d 6668	. . 3 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x H y ) = ( X H Y ) )
16	13, 14	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x ( 2nd ` F ) y ) = ( X ( 2nd ` F ) Y ) )
17	16	fveq1d 6193	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( x ( 2nd ` F ) y ) ` h ) = ( ( X ( 2nd ` F ) Y ) ` h ) )
18	13, 14	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x ( 2nd ` G ) y ) = ( X ( 2nd ` G ) Y ) )
19	18	fveq1d 6193	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( x ( 2nd ` G ) y ) ` h ) = ( ( X ( 2nd ` G ) Y ) ` h ) )
20	17, 19	opeq12d 4410	. . 3 |- ( ( 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 ) >. )
21	15, 20	mpteq12dv 4733	. 2 |- ( ( 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 ) >. ) )
22		prf1.x	. 2 |- ( ph -> X e. B )
23		prf2.y	. 2 |- ( ph -> Y e. B )
24		ovex 6678	. . . 4 |- ( X H Y ) e. _V
25	24	mptex 6486	. . 3 |- ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) e. _V
26	25	a1i 11	. 2 |- ( ph -> ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) e. _V )
27	12, 21, 22, 23, 26	ovmpt2d 6788	1 |- ( ph -> ( X ( 2nd ` P ) Y ) = ( h e. ( X H Y ) |-> <. ( ( X ( 2nd ` F ) Y ) ` h ) , ( ( X ( 2nd ` G ) Y ) ` h ) >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Func cfunc 16514   ⟨,⟩_F cprf 16811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-func 16518  df-prf 16815

This theorem is referenced by:  prf2  16842