Theorem comfeq 16366

Description: Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfeq.1	|- .x. = (comp ` C )
comfeq.2	|- .xb = (comp ` D )
comfeq.h	|- H = ( Hom ` C )
comfeq.3	|- ( ph -> B = ( Base ` C ) )
comfeq.4	|- ( ph -> B = ( Base ` D ) )
comfeq.5	|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )

Assertion

Ref	Expression
comfeq	|- ( ph -> ( (compf ` C ) = (compf ` D ) <-> A. x e. B A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) )

Distinct variable groups:   f, g, x, y, z, B   C, f, g, z   ph, f, g, z   .x. , f, g, x, y   D, f, g, z   f, H, g, x, y   .xb , f, g, x, y

Allowed substitution hints:   ph( x, y)   C( x, y)   D( x, y)   .xb ( z)   .x. ( z)   H( z)

Proof of Theorem comfeq

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . . . . 6 |- ( ( 2nd ` u ) H z ) e. _V
2		fvex 6201	. . . . . 6 |- ( H ` u ) e. _V
3	1, 2	mpt2ex 7247	. . . . 5 |- ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) e. _V
4	3	rgen2w 2925	. . . 4 |- A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) e. _V
5		mpt22eqb 6769	. . . 4 |- ( A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) e. _V -> ( ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) <-> A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) )
6	4, 5	ax-mp 5	. . 3 |- ( ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) <-> A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) )
7		vex 3203	. . . . . . . . 9 |- x e. _V
8		vex 3203	. . . . . . . . 9 |- y e. _V
9	7, 8	op2ndd 7179	. . . . . . . 8 |- ( u = <. x , y >. -> ( 2nd ` u ) = y )
10	9	oveq1d 6665	. . . . . . 7 |- ( u = <. x , y >. -> ( ( 2nd ` u ) H z ) = ( y H z ) )
11		fveq2 6191	. . . . . . . . 9 |- ( u = <. x , y >. -> ( H ` u ) = ( H ` <. x , y >. ) )
12		df-ov 6653	. . . . . . . . 9 |- ( x H y ) = ( H ` <. x , y >. )
13	11, 12	syl6eqr 2674	. . . . . . . 8 |- ( u = <. x , y >. -> ( H ` u ) = ( x H y ) )
14		oveq1 6657	. . . . . . . . . 10 |- ( u = <. x , y >. -> ( u .x. z ) = ( <. x , y >. .x. z ) )
15	14	oveqd 6667	. . . . . . . . 9 |- ( u = <. x , y >. -> ( g ( u .x. z ) f ) = ( g ( <. x , y >. .x. z ) f ) )
16		oveq1 6657	. . . . . . . . . 10 |- ( u = <. x , y >. -> ( u .xb z ) = ( <. x , y >. .xb z ) )
17	16	oveqd 6667	. . . . . . . . 9 |- ( u = <. x , y >. -> ( g ( u .xb z ) f ) = ( g ( <. x , y >. .xb z ) f ) )
18	15, 17	eqeq12d 2637	. . . . . . . 8 |- ( u = <. x , y >. -> ( ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) <-> ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) )
19	13, 18	raleqbidv 3152	. . . . . . 7 |- ( u = <. x , y >. -> ( A. f e. ( H ` u ) ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) <-> A. f e. ( x H y ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) )
20	10, 19	raleqbidv 3152	. . . . . 6 |- ( u = <. x , y >. -> ( A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) <-> A. g e. ( y H z ) A. f e. ( x H y ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) )
21		ovex 6678	. . . . . . . 8 |- ( g ( u .x. z ) f ) e. _V
22	21	rgen2w 2925	. . . . . . 7 |- A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) e. _V
23		mpt22eqb 6769	. . . . . . 7 |- ( A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) e. _V -> ( ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) ) )
24	22, 23	ax-mp 5	. . . . . 6 |- ( ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. g e. ( ( 2nd ` u ) H z ) A. f e. ( H ` u ) ( g ( u .x. z ) f ) = ( g ( u .xb z ) f ) )
25		ralcom 3098	. . . . . 6 |- ( A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) <-> A. g e. ( y H z ) A. f e. ( x H y ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) )
26	20, 24, 25	3bitr4g 303	. . . . 5 |- ( u = <. x , y >. -> ( ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) )
27	26	ralbidv 2986	. . . 4 |- ( u = <. x , y >. -> ( A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) )
28	27	ralxp 5263	. . 3 |- ( A. u e. ( B X. B ) A. z e. B ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) <-> A. x e. B A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) )
29	6, 28	bitri 264	. 2 |- ( ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) <-> A. x e. B A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) )
30		comfeq.3	. . . . . 6 |- ( ph -> B = ( Base ` C ) )
31	30	sqxpeqd 5141	. . . . 5 |- ( ph -> ( B X. B ) = ( ( Base ` C ) X. ( Base ` C ) ) )
32		eqidd 2623	. . . . 5 |- ( ph -> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) = ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) )
33	31, 30, 32	mpt2eq123dv 6717	. . . 4 |- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( ( Base ` C ) X. ( Base ` C ) ) , z e. ( Base ` C ) |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) )
34		eqid 2622	. . . . 5 |- (compf ` C ) = (compf ` C )
35		eqid 2622	. . . . 5 |- ( Base ` C ) = ( Base ` C )
36		comfeq.h	. . . . 5 |- H = ( Hom ` C )
37		comfeq.1	. . . . 5 |- .x. = (comp ` C )
38	34, 35, 36, 37	comfffval 16358	. . . 4 |- (compf ` C ) = ( u e. ( ( Base ` C ) X. ( Base ` C ) ) , z e. ( Base ` C ) |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) )
39	33, 38	syl6eqr 2674	. . 3 |- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = (compf ` C ) )
40		eqid 2622	. . . . . . . 8 |- ( Hom ` D ) = ( Hom ` D )
41		comfeq.5	. . . . . . . . 9 |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
42	41	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( Hom f ` C ) = ( Hom f ` D ) )
43		xp2nd 7199	. . . . . . . . . 10 |- ( u e. ( B X. B ) -> ( 2nd ` u ) e. B )
44	43	3ad2ant2 1083	. . . . . . . . 9 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( 2nd ` u ) e. B )
45	30	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> B = ( Base ` C ) )
46	44, 45	eleqtrd 2703	. . . . . . . 8 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( 2nd ` u ) e. ( Base ` C ) )
47		simp3 1063	. . . . . . . . 9 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> z e. B )
48	47, 45	eleqtrd 2703	. . . . . . . 8 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> z e. ( Base ` C ) )
49	35, 36, 40, 42, 46, 48	homfeqval 16357	. . . . . . 7 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( ( 2nd ` u ) H z ) = ( ( 2nd ` u ) ( Hom ` D ) z ) )
50		xp1st 7198	. . . . . . . . . . . 12 |- ( u e. ( B X. B ) -> ( 1st ` u ) e. B )
51	50	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( 1st ` u ) e. B )
52	51, 45	eleqtrd 2703	. . . . . . . . . 10 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( 1st ` u ) e. ( Base ` C ) )
53	35, 36, 40, 42, 52, 46	homfeqval 16357	. . . . . . . . 9 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( ( 1st ` u ) H ( 2nd ` u ) ) = ( ( 1st ` u ) ( Hom ` D ) ( 2nd ` u ) ) )
54		df-ov 6653	. . . . . . . . 9 |- ( ( 1st ` u ) H ( 2nd ` u ) ) = ( H ` <. ( 1st ` u ) , ( 2nd ` u ) >. )
55		df-ov 6653	. . . . . . . . 9 |- ( ( 1st ` u ) ( Hom ` D ) ( 2nd ` u ) ) = ( ( Hom ` D ) ` <. ( 1st ` u ) , ( 2nd ` u ) >. )
56	53, 54, 55	3eqtr3g 2679	. . . . . . . 8 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( H ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) = ( ( Hom ` D ) ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) )
57		1st2nd2 7205	. . . . . . . . . 10 |- ( u e. ( B X. B ) -> u = <. ( 1st ` u ) , ( 2nd ` u ) >. )
58	57	3ad2ant2 1083	. . . . . . . . 9 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> u = <. ( 1st ` u ) , ( 2nd ` u ) >. )
59	58	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( H ` u ) = ( H ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) )
60	58	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( ( Hom ` D ) ` u ) = ( ( Hom ` D ) ` <. ( 1st ` u ) , ( 2nd ` u ) >. ) )
61	56, 59, 60	3eqtr4d 2666	. . . . . . 7 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( H ` u ) = ( ( Hom ` D ) ` u ) )
62		eqidd 2623	. . . . . . 7 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( g ( u .xb z ) f ) = ( g ( u .xb z ) f ) )
63	49, 61, 62	mpt2eq123dv 6717	. . . . . 6 |- ( ( ph /\ u e. ( B X. B ) /\ z e. B ) -> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) = ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) )
64	63	mpt2eq3dva 6719	. . . . 5 |- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) )
65		comfeq.4	. . . . . . 7 |- ( ph -> B = ( Base ` D ) )
66	65	sqxpeqd 5141	. . . . . 6 |- ( ph -> ( B X. B ) = ( ( Base ` D ) X. ( Base ` D ) ) )
67		eqidd 2623	. . . . . 6 |- ( ph -> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) = ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) )
68	66, 65, 67	mpt2eq123dv 6717	. . . . 5 |- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) = ( u e. ( ( Base ` D ) X. ( Base ` D ) ) , z e. ( Base ` D ) |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) )
69	64, 68	eqtrd 2656	. . . 4 |- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) = ( u e. ( ( Base ` D ) X. ( Base ` D ) ) , z e. ( Base ` D ) |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) ) )
70		eqid 2622	. . . . 5 |- (compf ` D ) = (compf ` D )
71		eqid 2622	. . . . 5 |- ( Base ` D ) = ( Base ` D )
72		comfeq.2	. . . . 5 |- .xb = (comp ` D )
73	70, 71, 40, 72	comfffval 16358	. . . 4 |- (compf ` D ) = ( u e. ( ( Base ` D ) X. ( Base ` D ) ) , z e. ( Base ` D ) |-> ( g e. ( ( 2nd ` u ) ( Hom ` D ) z ) , f e. ( ( Hom ` D ) ` u ) |-> ( g ( u .xb z ) f ) ) )
74	69, 73	syl6eqr 2674	. . 3 |- ( ph -> ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) = (compf ` D ) )
75	39, 74	eqeq12d 2637	. 2 |- ( ph -> ( ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .x. z ) f ) ) ) = ( u e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` u ) H z ) , f e. ( H ` u ) |-> ( g ( u .xb z ) f ) ) ) <-> (compf ` C ) = (compf ` D ) ) )
76	29, 75	syl5rbbr 275	1 |- ( ph -> ( (compf ` C ) = (compf ` D ) <-> A. x e. B A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Hom _f chomf 16327  comp_fccomf 16328

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-homf 16331  df-comf 16332

This theorem is referenced by:  comfeqd  16367  2oppccomf  16385  oppccomfpropd  16387  resssetc  16742  resscatc  16755