Theorem rescabs 17791

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 2737	. . . 4 ⊢ (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽)
2		ovexd 7395	. . . 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 5261	. . . 4 ⊢ (𝜑 → 𝑇 ∈ V)
6		rescabs.j	. . . 4 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	1, 2, 5, 6	rescval2 17786	. . 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 7395	. . . . . . 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 2737	. . . . . . . 8 ⊢ (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇)
12		baseid 17173	. . . . . . . . 9 ⊢ Base = Slot (Base‘ndx)
13		slotsbhcdif 17369	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx))
14	13	simp1i 1140	. . . . . . . . 9 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
15	12, 14	setsnid 17169	. . . . . . . 8 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
16	11, 15	ressid2 17195	. . . . . . 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 1374	. . . . . 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 7375	. . . . 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 7393	. . . . . 6 ⊢ (𝐶 ↾s 𝑆) ∈ V
20	5, 5	xpexd 7698	. . . . . . . 8 ⊢ (𝜑 → (𝑇 × 𝑇) ∈ V)
21	6, 20	fnexd 7166	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
23		setsabs 17140	. . . . . 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 588	. . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆)
26		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
27	25, 26	ressbas 17197	. . . . . . . . . . . . 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 3954	. . . . . . . . . . 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 4179	. . . . . . . . . . 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 4182	. . . . . . . . 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 4185	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶)))
36	33, 35	eqssd 3940	. . . . . . . 8 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) = (𝑇 ∩ (Base‘𝐶)))
37	36	oveq2d 7376	. . . . . . 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 17206	. . . . . . . 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 17206	. . . . . . . 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 2782	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑇))
44	43	oveq1d 7375	. . . . 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 2776	. . . 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 7395	. . . . . . . 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 17196	. . . . . . . 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 1374	. . . . . . 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 7395	. . . . . . . 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 7698	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
56	54, 55	fnexd 7166	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ V)
57	56	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐻 ∈ V)
58		fvex 6847	. . . . . . . . . 10 ⊢ (Base‘(𝐶 ↾s 𝑆)) ∈ V
59	58	inex2 5255	. . . . . . . . 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 6847	. . . . . . . . 9 ⊢ (Hom ‘ndx) ∈ V
62		fvex 6847	. . . . . . . . 9 ⊢ (Base‘ndx) ∈ V
63	61, 62	setscom 17141	. . . . . . . 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 839	. . . . . . 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 2737	. . . . . . . . . . 11 ⊢ ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) ↾s 𝑇)
66		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘(𝐶 ↾s 𝑆))
67	65, 66	ressval2 17196	. . . . . . . . . 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 1374	. . . . . . . . 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 17209	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑇 ⊆ 𝑆) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇))
71	3, 69, 70	syl2an2r 686	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇))
72	68, 71	eqtr3d 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) = (𝐶 ↾s 𝑇))
73	72	oveq1d 7375	. . . . . . 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 2776	. . . . . 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 7375	. . . . 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 7393	. . . . . 6 ⊢ (𝐶 ↾s 𝑇) ∈ V
77	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
78		setsabs 17140	. . . . . 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 588	. . . . 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 2772	. . . 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 813	. . 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 2772	. 2 ⊢ (𝜑 → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
83		eqid 2737	. . . 4 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
84		rescabs.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
85	83, 84, 3, 54	rescval2 17786	. . 3 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
86	85	oveq1d 7375	. 2 ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽))
87		eqid 2737	. . 3 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
88	87, 84, 5, 6	rescval2 17786	. 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
89	82, 86, 88	3eqtr4d 2782	1 ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ⟨cop 4574   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360   sSet csts 17124  ndxcnx 17154  Basecbs 17170   ↾_s cress 17191  Hom chom 17222  compcco 17223   ↾_cat cresc 17766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-hom 17235  df-cco 17236  df-resc 17769

This theorem is referenced by:  subsubc  17811  fldc  20752  fldcALTV  48820