Theorem resscatc 18063

Description: The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
resscatc.c	⊢ 𝐶 = (CatCat‘𝑈)
resscatc.d	⊢ 𝐷 = (CatCat‘𝑉)
resscatc.1	⊢ (𝜑 → 𝑈 ∈ 𝑊)
resscatc.2	⊢ (𝜑 → 𝑉 ⊆ 𝑈)

Assertion

Ref	Expression
resscatc	⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))

Proof of Theorem resscatc

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

Step	Hyp	Ref	Expression
1		resscatc.d	. . . . . 6 ⊢ 𝐷 = (CatCat‘𝑉)
2		eqid 2730	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		resscatc.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
4		resscatc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
5	3, 4	ssexd 5323	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V)
6	5	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑉 ∈ V)
7		eqid 2730	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
8		simprl 767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (𝑉 ∩ Cat))
9	1, 2, 5	catcbas 18055	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐷) = (𝑉 ∩ Cat))
10	9	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (Base‘𝐷) = (𝑉 ∩ Cat))
11	8, 10	eleqtrrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐷))
12		simprr 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (𝑉 ∩ Cat))
13	12, 10	eleqtrrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐷))
14	1, 2, 6, 7, 11, 13	catchom 18057	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥 Func 𝑦))
15		resscatc.c	. . . . . 6 ⊢ 𝐶 = (CatCat‘𝑈)
16		eqid 2730	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
17	3	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑈 ∈ 𝑊)
18		eqid 2730	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19		inass 4218	. . . . . . . . . . 11 ⊢ ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ (𝑈 ∩ Cat))
20	15, 16, 3	catcbas 18055	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Cat))
21	20	ineq2d 4211	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∩ (Base‘𝐶)) = (𝑉 ∩ (𝑈 ∩ Cat)))
22	19, 21	eqtr4id 2789	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ (Base‘𝐶)))
23		df-ss 3964	. . . . . . . . . . . 12 ⊢ (𝑉 ⊆ 𝑈 ↔ (𝑉 ∩ 𝑈) = 𝑉)
24	4, 23	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = 𝑉)
25	24	ineq1d 4210	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ Cat))
26		eqid 2730	. . . . . . . . . . . 12 ⊢ (𝐶 ↾s 𝑉) = (𝐶 ↾s 𝑉)
27	26, 16	ressbas 17183	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (𝑉 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑉)))
28	5, 27	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑉)))
29	22, 25, 28	3eqtr3d 2778	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∩ Cat) = (Base‘(𝐶 ↾s 𝑉)))
30	26, 16	ressbasss 17187	. . . . . . . . 9 ⊢ (Base‘(𝐶 ↾s 𝑉)) ⊆ (Base‘𝐶)
31	29, 30	eqsstrdi 4035	. . . . . . . 8 ⊢ (𝜑 → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
32	31	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
33	32, 8	sseldd 3982	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐶))
34	32, 12	sseldd 3982	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐶))
35	15, 16, 17, 18, 33, 34	catchom 18057	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 Func 𝑦))
36	26, 18	resshom 17368	. . . . . . 7 ⊢ (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶 ↾s 𝑉)))
37	5, 36	syl 17	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘(𝐶 ↾s 𝑉)))
38	37	oveqdr 7439	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦))
39	14, 35, 38	3eqtr2rd 2777	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
40	39	ralrimivva 3198	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)(𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
41		eqid 2730	. . . 4 ⊢ (Hom ‘(𝐶 ↾s 𝑉)) = (Hom ‘(𝐶 ↾s 𝑉))
42	9	eqcomd 2736	. . . 4 ⊢ (𝜑 → (𝑉 ∩ Cat) = (Base‘𝐷))
43	41, 7, 29, 42	homfeq 17642	. . 3 ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)(𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
44	40, 43	mpbird 256	. 2 ⊢ (𝜑 → (Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷))
45	5	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
46		eqid 2730	. . . . . . . 8 ⊢ (comp‘𝐷) = (comp‘𝐷)
47		simplr1 1213	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ (𝑉 ∩ Cat))
48	9	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (Base‘𝐷) = (𝑉 ∩ Cat))
49	47, 48	eleqtrrd 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ (Base‘𝐷))
50		simplr2 1214	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ (𝑉 ∩ Cat))
51	50, 48	eleqtrrd 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ (Base‘𝐷))
52		simplr3 1215	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ (𝑉 ∩ Cat))
53	52, 48	eleqtrrd 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ (Base‘𝐷))
54		simprl 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
55	1, 2, 45, 7, 49, 51	catchom 18057	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥 Func 𝑦))
56	54, 55	eleqtrd 2833	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥 Func 𝑦))
57		simprr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
58	1, 2, 45, 7, 51, 53	catchom 18057	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑦(Hom ‘𝐷)𝑧) = (𝑦 Func 𝑧))
59	57, 58	eleqtrd 2833	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦 Func 𝑧))
60	1, 2, 45, 46, 49, 51, 53, 56, 59	catcco 18059	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔 ∘func 𝑓))
61	3	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈 ∈ 𝑊)
62		eqid 2730	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
63	31	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
64	63, 47	sseldd 3982	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ (Base‘𝐶))
65	63, 50	sseldd 3982	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ (Base‘𝐶))
66	63, 52	sseldd 3982	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ (Base‘𝐶))
67	15, 16, 61, 62, 64, 65, 66, 56, 59	catcco 18059	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔 ∘func 𝑓))
68	26, 62	ressco 17369	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶 ↾s 𝑉)))
69	5, 68	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (comp‘𝐶) = (comp‘(𝐶 ↾s 𝑉)))
70	69	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (comp‘𝐶) = (comp‘(𝐶 ↾s 𝑉)))
71	70	oveqd 7428	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧))
72	71	oveqd 7428	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓))
73	60, 67, 72	3eqtr2d 2776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓))
74	73	ralrimivva 3198	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓))
75	74	ralrimivvva 3201	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)∀𝑧 ∈ (𝑉 ∩ Cat)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓))
76		eqid 2730	. . . . 5 ⊢ (comp‘(𝐶 ↾s 𝑉)) = (comp‘(𝐶 ↾s 𝑉))
77	44	eqcomd 2736	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s 𝑉)))
78	46, 76, 7, 42, 29, 77	comfeq 17654	. . . 4 ⊢ (𝜑 → ((compf‘𝐷) = (compf‘(𝐶 ↾s 𝑉)) ↔ ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)∀𝑧 ∈ (𝑉 ∩ Cat)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶 ↾s 𝑉))𝑧)𝑓)))
79	75, 78	mpbird 256	. . 3 ⊢ (𝜑 → (compf‘𝐷) = (compf‘(𝐶 ↾s 𝑉)))
80	79	eqcomd 2736	. 2 ⊢ (𝜑 → (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷))
81	44, 80	jca 510	1 ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  Vcvv 3472   ∩ cin 3946   ⊆ wss 3947  ⟨cop 4633  ‘cfv 6542  (class class class)co 7411  Basecbs 17148   ↾_s cress 17177  Hom chom 17212  compcco 17213  Catccat 17612  Hom_f chomf 17614  comp_fccomf 17615   Func cfunc 17808   ∘_func ccofu 17810  CatCatccatc 18052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-hom 17225  df-cco 17226  df-homf 17618  df-comf 17619  df-catc 18053

This theorem is referenced by: (None)