hofpropd - Metamath Proof Explorer

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Theorem hofpropd 16907

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)

**Hypotheses**
Ref	Expression
hofpropd.1	_f _f
hofpropd.2	comp_f comp_f
hofpropd.c
hofpropd.d

**Assertion**
Ref	Expression
hofpropd	Hom_F Hom_F

Proof of Theorem hofpropd Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hofpropd.1	. . 3 _f _f
2	1	homfeqbas 16356	. . . . 5
3	2	sqxpeqd 5141	. . . 4
4	3	adantr 481	. . . 4 $/\$
5		eqid 2622	. . . . . 6
6		eqid 2622	. . . . . 6
7		eqid 2622	. . . . . 6
8	1	adantr 481	. . . . . 6 $/\$ $/\$ _f _f
9		xp1st 7198	. . . . . . 7
10	9	ad2antll 765	. . . . . 6 $/\$ $/\$
11		xp1st 7198	. . . . . . 7
12	11	ad2antrl 764	. . . . . 6 $/\$ $/\$
13	5, 6, 7, 8, 10, 12	homfeqval 16357	. . . . 5 $/\$ $/\$
14		xp2nd 7199	. . . . . . . 8
15	14	ad2antrl 764	. . . . . . 7 $/\$ $/\$
16		xp2nd 7199	. . . . . . . 8
17	16	ad2antll 765	. . . . . . 7 $/\$ $/\$
18	5, 6, 7, 8, 15, 17	homfeqval 16357	. . . . . 6 $/\$ $/\$
19	18	adantr 481	. . . . 5 $/\$ $/\$ $/\$
20	5, 6, 7, 8, 12, 15	homfeqval 16357	. . . . . . . . 9 $/\$ $/\$
21		df-ov 6653	. . . . . . . . 9
22		df-ov 6653	. . . . . . . . 9
23	20, 21, 22	3eqtr3g 2679	. . . . . . . 8 $/\$ $/\$
24		1st2nd2 7205	. . . . . . . . . 10
25	24	ad2antrl 764	. . . . . . . . 9 $/\$ $/\$
26	25	fveq2d 6195	. . . . . . . 8 $/\$ $/\$
27	25	fveq2d 6195	. . . . . . . 8 $/\$ $/\$
28	23, 26, 27	3eqtr4d 2666	. . . . . . 7 $/\$ $/\$
29	28	adantr 481	. . . . . 6 $/\$ $/\$ $/\$ $/\$
30		eqid 2622	. . . . . . . 8 comp comp
31		eqid 2622	. . . . . . . 8 comp comp
32	8	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ _f _f
33		hofpropd.2	. . . . . . . . 9 comp_f comp_f
34	33	ad3antrrr 766	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ comp_f comp_f
35	10	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
36	12	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
37	17	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
38		simplrl 800	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
39	25	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
40	39	oveq1d 6665	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp
41	40	oveqd 6667	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp
42		hofpropd.c	. . . . . . . . . . 11
43	42	ad3antrrr 766	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
44	15	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
45	26	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
46	45, 21	syl6eqr 2674	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
47	46	eleq2d 2687	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
48	47	biimpa 501	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
49		simplrr 801	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
50	5, 6, 30, 43, 36, 44, 37, 48, 49	catcocl 16346	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ comp
51	41, 50	eqeltrd 2701	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ comp
52	5, 6, 30, 31, 32, 34, 35, 36, 37, 38, 51	comfeqval 16368	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp comp comp
53	5, 6, 30, 31, 32, 34, 36, 44, 37, 48, 49	comfeqval 16368	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp
54	39	oveq1d 6665	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp
55	54	oveqd 6667	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp
56	53, 41, 55	3eqtr4d 2666	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp
57	56	oveq1d 6665	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp comp comp
58	52, 57	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ comp comp comp comp
59	29, 58	mpteq12dva 4732	. . . . 5 $/\$ $/\$ $/\$ $/\$ comp comp comp comp
60	13, 19, 59	mpt2eq123dva 6716	. . . 4 $/\$ $/\$ comp comp comp comp
61	3, 4, 60	mpt2eq123dva 6716	. . 3 comp comp comp comp
62	1, 61	opeq12d 4410	. 2 _f comp comp _f comp comp
63		eqid 2622	. . 3 Hom_F Hom_F
64	63, 42, 5, 6, 30	hofval 16892	. 2 Hom_F _f comp comp
65		eqid 2622	. . 3 Hom_F Hom_F
66		hofpropd.d	. . 3
67		eqid 2622	. . 3
68	65, 66, 67, 7, 31	hofval 16892	. 2 Hom_F _f comp comp
69	62, 64, 68	3eqtr4d 2666	1 Hom_F Hom_F

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384

wceq 1483

wcel 1990

cop 4183

cmpt 4729

cxp 5112

cfv 5888 (class class class)co 6650

cmpt2 6652

c1st 7166

c2nd 7167

cbs 15857

chom 15952 compcco 15953

ccat 16325

_f chomf 16327 comp_fccomf 16328 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-cat 16329 df-homf 16331 df-comf 16332 df-hof 16890

This theorem is referenced by: yonpropd 16908 oppcyon 16909

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