Theorem resscatc 18166

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 2769	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		resscatc.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
4		resscatc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
5	3, 4	ssexd 5295	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V)
6	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑉 ∈ V)
7		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
8		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (𝑉 ∩ Cat))
9	1, 2, 5	catcbas 18158	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐷) = (𝑉 ∩ Cat))
10	9	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (Base‘𝐷) = (𝑉 ∩ Cat))
11	8, 10	eleqtrrd 2872	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐷))
12		simprr 784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (𝑉 ∩ Cat))
13	12, 10	eleqtrrd 2872	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐷))
14	1, 2, 6, 7, 11, 13	catchom 18160	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥 Func 𝑦))
15		resscatc.c	. . . . . 6 ⊢ 𝐶 = (CatCat‘𝑈)
16		eqid 2769	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
17	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑈 ∈ 𝑊)
18		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19		inass 4188	. . . . . . . . . . 11 ⊢ ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ (𝑈 ∩ Cat))
20	15, 16, 3	catcbas 18158	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Cat))
21	20	ineq2d 4181	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∩ (Base‘𝐶)) = (𝑉 ∩ (𝑈 ∩ Cat)))
22	19, 21	eqtr4id 2823	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ (Base‘𝐶)))
23		dfss2 3931	. . . . . . . . . . . 12 ⊢ (𝑉 ⊆ 𝑈 ↔ (𝑉 ∩ 𝑈) = 𝑉)
24	4, 23	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = 𝑉)
25	24	ineq1d 4180	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ Cat))
26		eqid 2769	. . . . . . . . . . . 12 ⊢ (𝐶 ↾s 𝑉) = (𝐶 ↾s 𝑉)
27	26, 16	ressbas 17296	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (𝑉 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑉)))
28	5, 27	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑉)))
29	22, 25, 28	3eqtr3d 2812	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∩ Cat) = (Base‘(𝐶 ↾s 𝑉)))
30	26, 16	ressbasss 17299	. . . . . . . . 9 ⊢ (Base‘(𝐶 ↾s 𝑉)) ⊆ (Base‘𝐶)
31	29, 30	eqsstrdi 3989	. . . . . . . 8 ⊢ (𝜑 → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
32	31	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
33	32, 8	sseldd 3946	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐶))
34	32, 12	sseldd 3946	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐶))
35	15, 16, 17, 18, 33, 34	catchom 18160	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 Func 𝑦))
36	26, 18	resshom 17471	. . . . . . 7 ⊢ (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶 ↾s 𝑉)))
37	5, 36	syl 18	. . . . . 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 2811	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
40	39	ralrimivva 3214	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)(𝑥(Hom ‘(𝐶 ↾s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
41		eqid 2769	. . . 4 ⊢ (Hom ‘(𝐶 ↾s 𝑉)) = (Hom ‘(𝐶 ↾s 𝑉))
42	9	eqcomd 2775	. . . 4 ⊢ (𝜑 → (𝑉 ∩ Cat) = (Base‘𝐷))
43	41, 7, 29, 42	homfeq 17750	. . 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 260	. 2 ⊢ (𝜑 → (Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷))
45	5	ad2antrr 738	. . . . . . . 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 2769	. . . . . . . 8 ⊢ (comp‘𝐷) = (comp‘𝐷)
47		simplr1 1232	. . . . . . . . 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 738	. . . . . . . . 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 2872	. . . . . . . 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 1233	. . . . . . . . 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 2872	. . . . . . . 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 1234	. . . . . . . . 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 2872	. . . . . . . 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 782	. . . . . . . . 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 18160	. . . . . . . . 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 2871	. . . . . . . 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 784	. . . . . . . . 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 18160	. . . . . . . . 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 2871	. . . . . . . 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 18162	. . . . . . 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 738	. . . . . . . 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 2769	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
63	31	ad2antrr 738	. . . . . . . . 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 3946	. . . . . . . 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 3946	. . . . . . . 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 3946	. . . . . . . 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 18162	. . . . . . 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 17472	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶 ↾s 𝑉)))
69	5, 68	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (comp‘𝐶) = (comp‘(𝐶 ↾s 𝑉)))
70	69	ad2antrr 738	. . . . . . . . 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 2810	. . . . . 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 3214	. . . . 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 3217	. . . 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 2769	. . . . 5 ⊢ (comp‘(𝐶 ↾s 𝑉)) = (comp‘(𝐶 ↾s 𝑉))
77	44	eqcomd 2775	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘(𝐶 ↾s 𝑉)))
78	46, 76, 7, 42, 29, 77	comfeq 17762	. . . 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 260	. . 3 ⊢ (𝜑 → (compf‘𝐷) = (compf‘(𝐶 ↾s 𝑉)))
80	79	eqcomd 2775	. 2 ⊢ (𝜑 → (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷))
81	44, 80	jca 520	1 ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ⟨cop 4600  ‘cfv 6537  (class class class)co 7411  Basecbs 17269   ↾_s cress 17290  Hom chom 17321  compcco 17322  Catccat 17720  Hom_f chomf 17722  comp_fccomf 17723   Func cfunc 17911   ∘_func ccofu 17913  CatCatccatc 18155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-hom 17334  df-cco 17335  df-homf 17726  df-comf 17727  df-catc 18156

This theorem is referenced by: (None)