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Theorem resscatc 18092

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 → ((Hom_f ‘(𝐶 ↾_s 𝑉)) = (Hom_f ‘𝐷) ∧ (comp_f‘(𝐶 ↾_s 𝑉)) = (comp_f‘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 2728	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		resscatc.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
4		resscatc.2	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
5	3, 4	ssexd 5319	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑉 ∈ V)
7		eqid 2728	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
8		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (𝑉 ∩ Cat))
9	1, 2, 5	catcbas 18084	. . . . . . . 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 2832	. . . . . 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 2832	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐷))
14	1, 2, 6, 7, 11, 13	catchom 18086	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥 Func 𝑦))
15		resscatc.c	. . . . . 6 ⊢ 𝐶 = (CatCat‘𝑈)
16		eqid 2728	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
17	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑈 ∈ 𝑊)
18		eqid 2728	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19		inass 4216	. . . . . . . . . . 11 ⊢ ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ (𝑈 ∩ Cat))
20	15, 16, 3	catcbas 18084	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Cat))
21	20	ineq2d 4209	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∩ (Base‘𝐶)) = (𝑉 ∩ (𝑈 ∩ Cat)))
22	19, 21	eqtr4id 2787	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ (Base‘𝐶)))
23		df-ss 3962	. . . . . . . . . . . 12 ⊢ (𝑉 ⊆ 𝑈 ↔ (𝑉 ∩ 𝑈) = 𝑉)
24	4, 23	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = 𝑉)
25	24	ineq1d 4208	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ Cat) = (𝑉 ∩ Cat))
26		eqid 2728	. . . . . . . . . . . 12 ⊢ (𝐶 ↾_s 𝑉) = (𝐶 ↾_s 𝑉)
27	26, 16	ressbas 17209	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∩ Cat) = (Base‘(𝐶 ↾_s 𝑉)))
30	26, 16	ressbasss 17213	. . . . . . . . 9 ⊢ (Base‘(𝐶 ↾_s 𝑉)) ⊆ (Base‘𝐶)
31	29, 30	eqsstrdi 4033	. . . . . . . 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 3980	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐶))
34	32, 12	sseldd 3980	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐶))
35	15, 16, 17, 18, 33, 34	catchom 18086	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 Func 𝑦))
36	26, 18	resshom 17394	. . . . . . 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 7443	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶 ↾_s 𝑉))𝑦))
39	14, 35, 38	3eqtr2rd 2775	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘(𝐶 ↾_s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
40	39	ralrimivva 3196	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)(𝑥(Hom ‘(𝐶 ↾_s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
41		eqid 2728	. . . 4 ⊢ (Hom ‘(𝐶 ↾_s 𝑉)) = (Hom ‘(𝐶 ↾_s 𝑉))
42	9	eqcomd 2734	. . . 4 ⊢ (𝜑 → (𝑉 ∩ Cat) = (Base‘𝐷))
43	41, 7, 29, 42	homfeq 17668	. . 3 ⊢ (𝜑 → ((Hom_f ‘(𝐶 ↾_s 𝑉)) = (Hom_f ‘𝐷) ↔ ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)(𝑥(Hom ‘(𝐶 ↾_s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
44	40, 43	mpbird 257	. 2 ⊢ (𝜑 → (Hom_f ‘(𝐶 ↾_s 𝑉)) = (Hom_f ‘𝐷))
45	5	ad2antrr 725	. . . . . . . 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 2728	. . . . . . . 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 725	. . . . . . . . 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 2832	. . . . . . . 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 2832	. . . . . . . 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 2832	. . . . . . . 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 18086	. . . . . . . . 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 2831	. . . . . . . 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 18086	. . . . . . . . 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 2831	. . . . . . . 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 18088	. . . . . . 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 725	. . . . . . . 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 2728	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
63	31	ad2antrr 725	. . . . . . . . 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 3980	. . . . . . . 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 3980	. . . . . . . 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 3980	. . . . . . . 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 18088	. . . . . . 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 17395	. . . . . . . . . . 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 725	. . . . . . . . 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 7432	. . . . . . . 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 7432	. . . . . . 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 2774	. . . . . 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 3196	. . . . 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 3199	. . . 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 2728	. . . . 5 ⊢ (comp‘(𝐶 ↾_s 𝑉)) = (comp‘(𝐶 ↾_s 𝑉))
77	44	eqcomd 2734	. . . . 5 ⊢ (𝜑 → (Hom_f ‘𝐷) = (Hom_f ‘(𝐶 ↾_s 𝑉)))
78	46, 76, 7, 42, 29, 77	comfeq 17680	. . . 4 ⊢ (𝜑 → ((comp_f‘𝐷) = (comp_f‘(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 → (comp_f‘𝐷) = (comp_f‘(𝐶 ↾_s 𝑉)))
80	79	eqcomd 2734	. 2 ⊢ (𝜑 → (comp_f‘(𝐶 ↾_s 𝑉)) = (comp_f‘𝐷))
81	44, 80	jca 511	1 ⊢ (𝜑 → ((Hom_f ‘(𝐶 ↾_s 𝑉)) = (Hom_f ‘𝐷) ∧ (comp_f‘(𝐶 ↾_s 𝑉)) = (comp_f‘𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 ∩ cin 3944 ⊆ wss 3945 ⟨cop 4631 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 ↾_s cress 17203 Hom chom 17238 compcco 17239 Catccat 17638 Hom_f chomf 17640 comp_fccomf 17641 Func cfunc 17834 ∘_func ccofu 17836 CatCatccatc 18081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-hom 17251 df-cco 17252 df-homf 17644 df-comf 17645 df-catc 18082

This theorem is referenced by: (None)

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