Theorem hofval 16892

Description: Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from (oppCat ` C ) X. C to SetCat, whose object part is the hom-function Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
hofval.m	|- M = (HomF ` C )
hofval.c	|- ( ph -> C e. Cat )
hofval.b	|- B = ( Base ` C )
hofval.h	|- H = ( Hom ` C )
hofval.o	|- .x. = (comp ` C )

Assertion

Ref	Expression
hofval	|- ( ph -> M = <. ( Hom f ` C ) , ( x e. ( B X. B ) , y e. ( B X. B ) |-> ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) ) >. )

Distinct variable groups:   f, g, h, x, y, B   ph, f, g, h, x, y   C, f, g, h, x, y   f, H, g, h, x, y   .x. , f, g, h, x, y

Allowed substitution hints:   M( x, y, f, g, h)

Proof of Theorem hofval

Dummy variables b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hofval.m	. 2 |- M = (HomF ` C )
2		df-hof 16890	. . . 4 |- HomF = ( c e. Cat |-> <. ( Hom f ` c ) , [_ ( Base ` c ) / b ]_ ( x e. ( b X. b ) , y e. ( b X. b ) |-> ( f e. ( ( 1st ` y ) ( Hom ` c ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` c ) ` x ) |-> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) ) ) ) >. )
3	2	a1i 11	. . 3 |- ( ph -> HomF = ( c e. Cat |-> <. ( Hom f ` c ) , [_ ( Base ` c ) / b ]_ ( x e. ( b X. b ) , y e. ( b X. b ) |-> ( f e. ( ( 1st ` y ) ( Hom ` c ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` c ) ` x ) |-> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) ) ) ) >. ) )
4		simpr 477	. . . . 5 |- ( ( ph /\ c = C ) -> c = C )
5	4	fveq2d 6195	. . . 4 |- ( ( ph /\ c = C ) -> ( Hom f ` c ) = ( Hom f ` C ) )
6		fvexd 6203	. . . . 5 |- ( ( ph /\ c = C ) -> ( Base ` c ) e. _V )
7	4	fveq2d 6195	. . . . . 6 |- ( ( ph /\ c = C ) -> ( Base ` c ) = ( Base ` C ) )
8		hofval.b	. . . . . 6 |- B = ( Base ` C )
9	7, 8	syl6eqr 2674	. . . . 5 |- ( ( ph /\ c = C ) -> ( Base ` c ) = B )
10		simpr 477	. . . . . . 7 |- ( ( ( ph /\ c = C ) /\ b = B ) -> b = B )
11	10	sqxpeqd 5141	. . . . . 6 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( b X. b ) = ( B X. B ) )
12		simplr 792	. . . . . . . . . 10 |- ( ( ( ph /\ c = C ) /\ b = B ) -> c = C )
13	12	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( Hom ` c ) = ( Hom ` C ) )
14		hofval.h	. . . . . . . . 9 |- H = ( Hom ` C )
15	13, 14	syl6eqr 2674	. . . . . . . 8 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( Hom ` c ) = H )
16	15	oveqd 6667	. . . . . . 7 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( ( 1st ` y ) ( Hom ` c ) ( 1st ` x ) ) = ( ( 1st ` y ) H ( 1st ` x ) ) )
17	15	oveqd 6667	. . . . . . 7 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) = ( ( 2nd ` x ) H ( 2nd ` y ) ) )
18	15	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( ( Hom ` c ) ` x ) = ( H ` x ) )
19	12	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ph /\ c = C ) /\ b = B ) -> (comp ` c ) = (comp ` C ) )
20		hofval.o	. . . . . . . . . . 11 |- .x. = (comp ` C )
21	19, 20	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( ph /\ c = C ) /\ b = B ) -> (comp ` c ) = .x. )
22	21	oveqd 6667	. . . . . . . . 9 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) = ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) )
23	21	oveqd 6667	. . . . . . . . . 10 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( x (comp ` c ) ( 2nd ` y ) ) = ( x .x. ( 2nd ` y ) ) )
24	23	oveqd 6667	. . . . . . . . 9 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) = ( g ( x .x. ( 2nd ` y ) ) h ) )
25		eqidd 2623	. . . . . . . . 9 |- ( ( ( ph /\ c = C ) /\ b = B ) -> f = f )
26	22, 24, 25	oveq123d 6671	. . . . . . . 8 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) = ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) )
27	18, 26	mpteq12dv 4733	. . . . . . 7 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( h e. ( ( Hom ` c ) ` x ) |-> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) ) = ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) )
28	16, 17, 27	mpt2eq123dv 6717	. . . . . 6 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( f e. ( ( 1st ` y ) ( Hom ` c ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` c ) ` x ) |-> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) ) ) = ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) )
29	11, 11, 28	mpt2eq123dv 6717	. . . . 5 |- ( ( ( ph /\ c = C ) /\ b = B ) -> ( x e. ( b X. b ) , y e. ( b X. b ) |-> ( f e. ( ( 1st ` y ) ( Hom ` c ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` c ) ` x ) |-> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) ) ) ) = ( x e. ( B X. B ) , y e. ( B X. B ) |-> ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) ) )
30	6, 9, 29	csbied2 3561	. . . 4 |- ( ( ph /\ c = C ) -> [_ ( Base ` c ) / b ]_ ( x e. ( b X. b ) , y e. ( b X. b ) |-> ( f e. ( ( 1st ` y ) ( Hom ` c ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` c ) ` x ) |-> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) ) ) ) = ( x e. ( B X. B ) , y e. ( B X. B ) |-> ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) ) )
31	5, 30	opeq12d 4410	. . 3 |- ( ( ph /\ c = C ) -> <. ( Hom f ` c ) , [_ ( Base ` c ) / b ]_ ( x e. ( b X. b ) , y e. ( b X. b ) |-> ( f e. ( ( 1st ` y ) ( Hom ` c ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` c ) ` x ) |-> ( ( g ( x (comp ` c ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` c ) ( 2nd ` y ) ) f ) ) ) ) >. = <. ( Hom f ` C ) , ( x e. ( B X. B ) , y e. ( B X. B ) |-> ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) ) >. )
32		hofval.c	. . 3 |- ( ph -> C e. Cat )
33		opex 4932	. . . 4 |- <. ( Hom f ` C ) , ( x e. ( B X. B ) , y e. ( B X. B ) |-> ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) ) >. e. _V
34	33	a1i 11	. . 3 |- ( ph -> <. ( Hom f ` C ) , ( x e. ( B X. B ) , y e. ( B X. B ) |-> ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) ) >. e. _V )
35	3, 31, 32, 34	fvmptd 6288	. 2 |- ( ph -> (HomF ` C ) = <. ( Hom f ` C ) , ( x e. ( B X. B ) , y e. ( B X. B ) |-> ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) ) >. )
36	1, 35	syl5eq 2668	1 |- ( ph -> M = <. ( Hom f ` C ) , ( x e. ( B X. B ) , y e. ( B X. B ) |-> ( f e. ( ( 1st ` y ) H ( 1st ` x ) ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Hom _f chomf 16327  Hom_Fchof 16888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-hof 16890

This theorem is referenced by:  hof1fval  16893  hof2fval  16895  hofcl  16899  hofpropd  16907