Theorem hof2fval 16895

Description: The morphism part of the Hom functor, for morphisms <. f , g >. : <. X , Y >. --> <. Z , W >. (which since the first argument is contravariant means morphisms f : Z --> X and g : Y --> W), yields a function (a morphism of SetCat) mapping h : X --> Y to g o. h o. f : Z --> W. (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
hofval.m	|- M = (HomF ` C )
hofval.c	|- ( ph -> C e. Cat )
hof1.b	|- B = ( Base ` C )
hof1.h	|- H = ( Hom ` C )
hof1.x	|- ( ph -> X e. B )
hof1.y	|- ( ph -> Y e. B )
hof2.z	|- ( ph -> Z e. B )
hof2.w	|- ( ph -> W e. B )
hof2.o	|- .x. = (comp ` C )

Assertion

Ref	Expression
hof2fval	|- ( ph -> ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) = ( f e. ( Z H X ) , g e. ( Y H W ) |-> ( h e. ( X H Y ) |-> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) ) ) )

Distinct variable groups:   f, g, h, B   ph, f, g, h   C, f, g, h   f, H, g, h   f, W, g, h   .x. , f, g, h   f, X, g, h   f, Y, g, h   f, Z, g, h

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

Proof of Theorem hof2fval

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

Step	Hyp	Ref	Expression
1		hofval.m	. . . 4 |- M = (HomF ` C )
2		hofval.c	. . . 4 |- ( ph -> C e. Cat )
3		hof1.b	. . . 4 |- B = ( Base ` C )
4		hof1.h	. . . 4 |- H = ( Hom ` C )
5		hof2.o	. . . 4 |- .x. = (comp ` C )
6	1, 2, 3, 4, 5	hofval 16892	. . 3 |- ( 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 ) ) ) ) >. )
7		fvex 6201	. . . 4 |- ( Hom f ` C ) e. _V
8		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
9	3, 8	eqeltri 2697	. . . . . 6 |- B e. _V
10	9, 9	xpex 6962	. . . . 5 |- ( B X. B ) e. _V
11	10, 10	mpt2ex 7247	. . . 4 |- ( 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
12	7, 11	op2ndd 7179	. . 3 |- ( 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 ) ) ) ) >. -> ( 2nd ` M ) = ( 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 ) ) ) ) )
13	6, 12	syl 17	. 2 |- ( ph -> ( 2nd ` M ) = ( 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 ) ) ) ) )
14		simprr 796	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> y = <. Z , W >. )
15	14	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 1st ` y ) = ( 1st ` <. Z , W >. ) )
16		hof2.z	. . . . . . 7 |- ( ph -> Z e. B )
17		hof2.w	. . . . . . 7 |- ( ph -> W e. B )
18		op1stg 7180	. . . . . . 7 |- ( ( Z e. B /\ W e. B ) -> ( 1st ` <. Z , W >. ) = Z )
19	16, 17, 18	syl2anc 693	. . . . . 6 |- ( ph -> ( 1st ` <. Z , W >. ) = Z )
20	19	adantr 481	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 1st ` <. Z , W >. ) = Z )
21	15, 20	eqtrd 2656	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 1st ` y ) = Z )
22		simprl 794	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> x = <. X , Y >. )
23	22	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 1st ` x ) = ( 1st ` <. X , Y >. ) )
24		hof1.x	. . . . . . 7 |- ( ph -> X e. B )
25		hof1.y	. . . . . . 7 |- ( ph -> Y e. B )
26		op1stg 7180	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> ( 1st ` <. X , Y >. ) = X )
27	24, 25, 26	syl2anc 693	. . . . . 6 |- ( ph -> ( 1st ` <. X , Y >. ) = X )
28	27	adantr 481	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 1st ` <. X , Y >. ) = X )
29	23, 28	eqtrd 2656	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 1st ` x ) = X )
30	21, 29	oveq12d 6668	. . 3 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( ( 1st ` y ) H ( 1st ` x ) ) = ( Z H X ) )
31	22	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 2nd ` x ) = ( 2nd ` <. X , Y >. ) )
32		op2ndg 7181	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> ( 2nd ` <. X , Y >. ) = Y )
33	24, 25, 32	syl2anc 693	. . . . . 6 |- ( ph -> ( 2nd ` <. X , Y >. ) = Y )
34	33	adantr 481	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 2nd ` <. X , Y >. ) = Y )
35	31, 34	eqtrd 2656	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 2nd ` x ) = Y )
36	14	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 2nd ` y ) = ( 2nd ` <. Z , W >. ) )
37		op2ndg 7181	. . . . . . 7 |- ( ( Z e. B /\ W e. B ) -> ( 2nd ` <. Z , W >. ) = W )
38	16, 17, 37	syl2anc 693	. . . . . 6 |- ( ph -> ( 2nd ` <. Z , W >. ) = W )
39	38	adantr 481	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 2nd ` <. Z , W >. ) = W )
40	36, 39	eqtrd 2656	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 2nd ` y ) = W )
41	35, 40	oveq12d 6668	. . 3 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( ( 2nd ` x ) H ( 2nd ` y ) ) = ( Y H W ) )
42	22	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( H ` x ) = ( H ` <. X , Y >. ) )
43		df-ov 6653	. . . . 5 |- ( X H Y ) = ( H ` <. X , Y >. )
44	42, 43	syl6eqr 2674	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( H ` x ) = ( X H Y ) )
45	21, 29	opeq12d 4410	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> <. ( 1st ` y ) , ( 1st ` x ) >. = <. Z , X >. )
46	45, 40	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) = ( <. Z , X >. .x. W ) )
47	22, 40	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( x .x. ( 2nd ` y ) ) = ( <. X , Y >. .x. W ) )
48	47	oveqd 6667	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( g ( x .x. ( 2nd ` y ) ) h ) = ( g ( <. X , Y >. .x. W ) h ) )
49		eqidd 2623	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> f = f )
50	46, 48, 49	oveq123d 6671	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) = ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) )
51	44, 50	mpteq12dv 4733	. . 3 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( h e. ( H ` x ) |-> ( ( g ( x .x. ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. .x. ( 2nd ` y ) ) f ) ) = ( h e. ( X H Y ) |-> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) ) )
52	30, 41, 51	mpt2eq123dv 6717	. 2 |- ( ( ph /\ ( x = <. X , Y >. /\ y = <. Z , W >. ) ) -> ( 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 ) ) ) = ( f e. ( Z H X ) , g e. ( Y H W ) |-> ( h e. ( X H Y ) |-> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) ) ) )
53		opelxpi 5148	. . 3 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
54	24, 25, 53	syl2anc 693	. 2 |- ( ph -> <. X , Y >. e. ( B X. B ) )
55		opelxpi 5148	. . 3 |- ( ( Z e. B /\ W e. B ) -> <. Z , W >. e. ( B X. B ) )
56	16, 17, 55	syl2anc 693	. 2 |- ( ph -> <. Z , W >. e. ( B X. B ) )
57		ovex 6678	. . . 4 |- ( Z H X ) e. _V
58		ovex 6678	. . . 4 |- ( Y H W ) e. _V
59	57, 58	mpt2ex 7247	. . 3 |- ( f e. ( Z H X ) , g e. ( Y H W ) |-> ( h e. ( X H Y ) |-> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) ) ) e. _V
60	59	a1i 11	. 2 |- ( ph -> ( f e. ( Z H X ) , g e. ( Y H W ) |-> ( h e. ( X H Y ) |-> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) ) ) e. _V )
61	13, 52, 54, 56, 60	ovmpt2d 6788	1 |- ( ph -> ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) = ( f e. ( Z H X ) , g e. ( Y H W ) |-> ( h e. ( X H Y ) |-> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.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-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-hof 16890

This theorem is referenced by:  hof2val  16896