Theorem rescabs 17547

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 2738	. . . 4 ⊢ (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽)
2		ovexd 7310	. . . 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 5248	. . . 4 ⊢ (𝜑 → 𝑇 ∈ V)
6		rescabs.j	. . . 4 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	1, 2, 5, 6	rescval2 17540	. . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇)
9		ovexd 7310	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
10	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
11		eqid 2738	. . . . . . . 8 ⊢ (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇)
12		baseid 16915	. . . . . . . . 9 ⊢ Base = Slot (Base‘ndx)
13		slotsbhcdif 17125	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx))
14	13	simp1i 1138	. . . . . . . . 9 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
15	12, 14	setsnid 16910	. . . . . . . 8 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
16	11, 15	ressid2 16945	. . . . . . 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 1370	. . . . . 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 7290	. . . . 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 7308	. . . . . 6 ⊢ (𝐶 ↾s 𝑆) ∈ V
20	5, 5	xpexd 7601	. . . . . . . 8 ⊢ (𝜑 → (𝑇 × 𝑇) ∈ V)
21	6, 20	fnexd 7094	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
22	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
23		setsabs 16880	. . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆)
26		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
27	25, 26	ressbas 16947	. . . . . . . . . . . . 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 3952	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇 ↔ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇))
30	29	biimpar 478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇)
31		inss2 4163	. . . . . . . . . . 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 4166	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (𝑇 ∩ (Base‘𝐶)))
34	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ⊆ 𝑆)
35	34	ssrind 4169	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶)))
36	33, 35	eqssd 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) = (𝑇 ∩ (Base‘𝐶)))
37	36	oveq2d 7291	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶))))
38	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑆 ∈ 𝑊)
39	26	ressinbas 16955	. . . . . . . 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 16955	. . . . . . . 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 2788	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑇))
44	43	oveq1d 7290	. . . . 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 2782	. . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇)
47		ovexd 7310	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
48	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
49	11, 15	ressval2 16946	. . . . . . . 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 1370	. . . . . . 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 7310	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) ∈ V)
52	14	necomi 2998	. . . . . . . . 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 7601	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
56	54, 55	fnexd 7094	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ V)
57	56	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐻 ∈ V)
58		fvex 6787	. . . . . . . . . 10 ⊢ (Base‘(𝐶 ↾s 𝑆)) ∈ V
59	58	inex2 5242	. . . . . . . . 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 6787	. . . . . . . . 9 ⊢ (Hom ‘ndx) ∈ V
62		fvex 6787	. . . . . . . . 9 ⊢ (Base‘ndx) ∈ V
63	61, 62	setscom 16881	. . . . . . . 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 836	. . . . . . 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 2738	. . . . . . . . . . 11 ⊢ ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) ↾s 𝑇)
66		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘(𝐶 ↾s 𝑆))
67	65, 66	ressval2 16946	. . . . . . . . . 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 1370	. . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ⊆ 𝑆)
70		ressabs 16959	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑇 ⊆ 𝑆) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇))
71	3, 69, 70	syl2an2r 682	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇))
72	68, 71	eqtr3d 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) = (𝐶 ↾s 𝑇))
73	72	oveq1d 7290	. . . . . . 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 2782	. . . . . 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 7290	. . . . 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 7308	. . . . . 6 ⊢ (𝐶 ↾s 𝑇) ∈ V
77	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
78		setsabs 16880	. . . . . 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 2778	. . . 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 810	. . 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 2778	. 2 ⊢ (𝜑 → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
83		eqid 2738	. . . 4 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
84		rescabs.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
85	83, 84, 3, 54	rescval2 17540	. . 3 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
86	85	oveq1d 7290	. 2 ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽))
87		eqid 2738	. . 3 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
88	87, 84, 5, 6	rescval2 17540	. 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
89	82, 86, 88	3eqtr4d 2788	1 ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ⟨cop 4567   × cxp 5587   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   sSet csts 16864  ndxcnx 16894  Basecbs 16912   ↾_s cress 16941  Hom chom 16973  compcco 16974   ↾_cat cresc 17520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-hom 16986  df-cco 16987  df-resc 17523

This theorem is referenced by:  subsubc  17568  fldc  45641  fldcALTV  45659