Theorem resscatc 18047

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 2729	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		resscatc.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
4		resscatc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
5	3, 4	ssexd 5274	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑉 ∈ V)
7		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
8		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (𝑉 ∩ Cat))
9	1, 2, 5	catcbas 18039	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐷) = (𝑉 ∩ Cat))
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (Base‘𝐷) = (𝑉 ∩ Cat))
11	8, 10	eleqtrrd 2831	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐷))
12		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (𝑉 ∩ Cat))
13	12, 10	eleqtrrd 2831	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐷))
14	1, 2, 6, 7, 11, 13	catchom 18041	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥 Func 𝑦))
15		resscatc.c	. . . . . 6 ⊢ 𝐶 = (CatCat‘𝑈)
16		eqid 2729	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
17	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑈 ∈ 𝑊)
18		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19		inass 4187	. . . . . . . . . . 11 ⊢ ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ (𝑈 ∩ Cat))
20	15, 16, 3	catcbas 18039	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Cat))
21	20	ineq2d 4179	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∩ (Base‘𝐶)) = (𝑉 ∩ (𝑈 ∩ Cat)))
22	19, 21	eqtr4id 2783	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ (Base‘𝐶)))
23		dfss2 3929	. . . . . . . . . . . 12 ⊢ (𝑉 ⊆ 𝑈 ↔ (𝑉 ∩ 𝑈) = 𝑉)
24	4, 23	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = 𝑉)
25	24	ineq1d 4178	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ Cat))
26		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝐶 ↾s 𝑉) = (𝐶 ↾s 𝑉)
27	26, 16	ressbas 17182	. . . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∩ Cat) = (Base‘(𝐶 ↾s 𝑉)))
30	26, 16	ressbasss 17185	. . . . . . . . 9 ⊢ (Base‘(𝐶 ↾s 𝑉)) ⊆ (Base‘𝐶)
31	29, 30	eqsstrdi 3988	. . . . . . . 8 ⊢ (𝜑 → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
32	31	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
33	32, 8	sseldd 3944	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐶))
34	32, 12	sseldd 3944	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐶))
35	15, 16, 17, 18, 33, 34	catchom 18041	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 Func 𝑦))
36	26, 18	resshom 17357	. . . . . . 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 7397	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦))
39	14, 35, 38	3eqtr2rd 2771	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
40	39	ralrimivva 3178	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)(𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
41		eqid 2729	. . . 4 ⊢ (Hom ‘(𝐶 ↾s 𝑉)) = (Hom ‘(𝐶 ↾s 𝑉))
42	9	eqcomd 2735	. . . 4 ⊢ (𝜑 → (𝑉 ∩ Cat) = (Base‘𝐷))
43	41, 7, 29, 42	homfeq 17631	. . 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 257	. 2 ⊢ (𝜑 → (Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷))
45	5	ad2antrr 726	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (comp‘𝐷) = (comp‘𝐷)
47		simplr1 1216	. . . . . . . . 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 726	. . . . . . . . 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 2831	. . . . . . . 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 1217	. . . . . . . . 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 2831	. . . . . . . 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 1218	. . . . . . . . 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 2831	. . . . . . . 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 770	. . . . . . . . 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 18041	. . . . . . . . 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 2830	. . . . . . . 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 772	. . . . . . . . 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 18041	. . . . . . . . 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 2830	. . . . . . . 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 18043	. . . . . . 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 726	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
63	31	ad2antrr 726	. . . . . . . . 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 3944	. . . . . . . 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 3944	. . . . . . . 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 3944	. . . . . . . 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 18043	. . . . . . 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 17358	. . . . . . . . . . 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 726	. . . . . . . . 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 7386	. . . . . . . 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 7386	. . . . . . 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 2770	. . . . . 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 3178	. . . . 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 3181	. . . 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 2729	. . . . 5 ⊢ (comp‘(𝐶 ↾s 𝑉)) = (comp‘(𝐶 ↾s 𝑉))
77	44	eqcomd 2735	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s 𝑉)))
78	46, 76, 7, 42, 29, 77	comfeq 17643	. . . 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 257	. . 3 ⊢ (𝜑 → (compf‘𝐷) = (compf‘(𝐶 ↾s 𝑉)))
80	79	eqcomd 2735	. 2 ⊢ (𝜑 → (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷))
81	44, 80	jca 511	1 ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  ⟨cop 4591  ‘cfv 6499  (class class class)co 7369  Basecbs 17155   ↾_s cress 17176  Hom chom 17207  compcco 17208  Catccat 17601  Hom_f chomf 17603  comp_fccomf 17604   Func cfunc 17792   ∘_func ccofu 17794  CatCatccatc 18036

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-hom 17220  df-cco 17221  df-homf 17607  df-comf 17608  df-catc 18037

This theorem is referenced by: (None)