Theorem rescabs 17851

Description: Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 9-Nov-2024.)

Hypotheses

Ref	Expression
rescabs.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
rescabs.h	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
rescabs.j	⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
rescabs.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
rescabs.t	⊢ (𝜑 → 𝑇 ⊆ 𝑆)

Assertion

Ref	Expression
rescabs	⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))

Proof of Theorem rescabs

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽)
2		ovexd 7445	. . . 4 ⊢ (𝜑 → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
3		rescabs.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
4		rescabs.t	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑆)
5	3, 4	ssexd 5299	. . . 4 ⊢ (𝜑 → 𝑇 ∈ V)
6		rescabs.j	. . . 4 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	1, 2, 5, 6	rescval2 17846	. . 3 ⊢ (𝜑 → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
8		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇)
9		ovexd 7445	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
10	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
11		eqid 2736	. . . . . . . 8 ⊢ (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇)
12		baseid 17236	. . . . . . . . 9 ⊢ Base = Slot (Base‘ndx)
13		slotsbhcdif 17434	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx))
14	13	simp1i 1139	. . . . . . . . 9 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
15	12, 14	setsnid 17232	. . . . . . . 8 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
16	11, 15	ressid2 17260	. . . . . . 7 ⊢ (((Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇 ∧ ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V ∧ 𝑇 ∈ V) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
17	8, 9, 10, 16	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
18	17	oveq1d 7425	. . . . 5 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩))
19		ovex 7443	. . . . . 6 ⊢ (𝐶 ↾s 𝑆) ∈ V
20	5, 5	xpexd 7750	. . . . . . . 8 ⊢ (𝜑 → (𝑇 × 𝑇) ∈ V)
21	6, 20	fnexd 7215	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
23		setsabs 17203	. . . . . 6 ⊢ (((𝐶 ↾s 𝑆) ∈ V ∧ 𝐽 ∈ V) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩))
24	19, 22, 23	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩))
25		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆)
26		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
27	25, 26	ressbas 17262	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑊 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑆)))
28	3, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑆)))
29	28	sseq1d 3995	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇 ↔ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇))
30	29	biimpar 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇)
31		inss2 4218	. . . . . . . . . . 11 ⊢ (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
32	31	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
33	30, 32	ssind 4221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (𝑇 ∩ (Base‘𝐶)))
34	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ⊆ 𝑆)
35	34	ssrind 4224	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶)))
36	33, 35	eqssd 3981	. . . . . . . 8 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) = (𝑇 ∩ (Base‘𝐶)))
37	36	oveq2d 7426	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶))))
38	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑆 ∈ 𝑊)
39	26	ressinbas 17271	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑊 → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
40	38, 39	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
41	26	ressinbas 17271	. . . . . . . 8 ⊢ (𝑇 ∈ V → (𝐶 ↾s 𝑇) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶))))
42	10, 41	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑇) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶))))
43	37, 40, 42	3eqtr4d 2781	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑇))
44	43	oveq1d 7425	. . . . 5 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
45	18, 24, 44	3eqtrd 2775	. . . 4 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
46		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇)
47		ovexd 7445	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
48	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
49	11, 15	ressval2 17261	. . . . . . . 8 ⊢ ((¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇 ∧ ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V ∧ 𝑇 ∈ V) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩))
50	46, 47, 48, 49	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩))
51		ovexd 7445	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) ∈ V)
52	14	necomi 2987	. . . . . . . . 9 ⊢ (Hom ‘ndx) ≠ (Base‘ndx)
53	52	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (Hom ‘ndx) ≠ (Base‘ndx))
54		rescabs.h	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
55	3, 3	xpexd 7750	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
56	54, 55	fnexd 7215	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ V)
57	56	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐻 ∈ V)
58		fvex 6894	. . . . . . . . . 10 ⊢ (Base‘(𝐶 ↾s 𝑆)) ∈ V
59	58	inex2 5293	. . . . . . . . 9 ⊢ (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆))) ∈ V
60	59	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆))) ∈ V)
61		fvex 6894	. . . . . . . . 9 ⊢ (Hom ‘ndx) ∈ V
62		fvex 6894	. . . . . . . . 9 ⊢ (Base‘ndx) ∈ V
63	61, 62	setscom 17204	. . . . . . . 8 ⊢ ((((𝐶 ↾s 𝑆) ∈ V ∧ (Hom ‘ndx) ≠ (Base‘ndx)) ∧ (𝐻 ∈ V ∧ (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆))) ∈ V)) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) = (((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) sSet ⟨(Hom ‘ndx), 𝐻⟩))
64	51, 53, 57, 60, 63	syl22anc 838	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) = (((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) sSet ⟨(Hom ‘ndx), 𝐻⟩))
65		eqid 2736	. . . . . . . . . . 11 ⊢ ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) ↾s 𝑇)
66		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘(𝐶 ↾s 𝑆))
67	65, 66	ressval2 17261	. . . . . . . . . 10 ⊢ ((¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇 ∧ (𝐶 ↾s 𝑆) ∈ V ∧ 𝑇 ∈ V) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩))
68	46, 51, 48, 67	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩))
69	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ⊆ 𝑆)
70		ressabs 17274	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑇 ⊆ 𝑆) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇))
71	3, 69, 70	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇))
72	68, 71	eqtr3d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) = (𝐶 ↾s 𝑇))
73	72	oveq1d 7425	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩))
74	50, 64, 73	3eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩))
75	74	oveq1d 7425	. . . . 5 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = (((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩))
76		ovex 7443	. . . . . 6 ⊢ (𝐶 ↾s 𝑇) ∈ V
77	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
78		setsabs 17203	. . . . . 6 ⊢ (((𝐶 ↾s 𝑇) ∈ V ∧ 𝐽 ∈ V) → (((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
79	76, 77, 78	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
80	75, 79	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
81	45, 80	pm2.61dan 812	. . 3 ⊢ (𝜑 → ((((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
82	7, 81	eqtrd 2771	. 2 ⊢ (𝜑 → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
83		eqid 2736	. . . 4 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
84		rescabs.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
85	83, 84, 3, 54	rescval2 17846	. . 3 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
86	85	oveq1d 7425	. 2 ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽))
87		eqid 2736	. . 3 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
88	87, 84, 5, 6	rescval2 17846	. 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
89	82, 86, 88	3eqtr4d 2781	1 ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931  ⟨cop 4612   × cxp 5657   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   sSet csts 17187  ndxcnx 17217  Basecbs 17233   ↾_s cress 17256  Hom chom 17287  compcco 17288   ↾_cat cresc 17826

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-hom 17300  df-cco 17301  df-resc 17829

This theorem is referenced by:  subsubc  17871  fldc  20749  fldcALTV  48274