Theorem rescabsOLD 17593

Description: Obsolete proof of seqp1d 13784 as of 10-Nov-2024. Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

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
rescabsOLD	⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))

Proof of Theorem rescabsOLD

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 7342	. . . 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 5257	. . . 4 ⊢ (𝜑 → 𝑇 ∈ V)
6		rescabs.j	. . . 4 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
7	1, 2, 5, 6	rescval2 17585	. . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇)
9		ovexd 7342	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
10	5	adantr 482	. . . . . . 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 16960	. . . . . . . . 9 ⊢ Base = Slot (Base‘ndx)
13		1re 11021	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
14		1nn 12030	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
15		4nn0 12298	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ0
16		1nn0 12295	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ0
17		1lt10 12622	. . . . . . . . . . . 12 ⊢ 1 < ;10
18	14, 15, 16, 17	declti 12521	. . . . . . . . . . 11 ⊢ 1 < ;14
19	13, 18	ltneii 11134	. . . . . . . . . 10 ⊢ 1 ≠ ;14
20		basendx 16966	. . . . . . . . . . 11 ⊢ (Base‘ndx) = 1
21		homndx 17166	. . . . . . . . . . 11 ⊢ (Hom ‘ndx) = ;14
22	20, 21	neeq12i 3008	. . . . . . . . . 10 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14)
23	19, 22	mpbir 230	. . . . . . . . 9 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
24	12, 23	setsnid 16955	. . . . . . . 8 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
25	11, 24	ressid2 16990	. . . . . . 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⟩))
26	8, 9, 10, 25	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
27	26	oveq1d 7322	. . . . 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⟩))
28		ovex 7340	. . . . . 6 ⊢ (𝐶 ↾s 𝑆) ∈ V
29	5, 5	xpexd 7633	. . . . . . . 8 ⊢ (𝜑 → (𝑇 × 𝑇) ∈ V)
30		fnex 7125	. . . . . . . 8 ⊢ ((𝐽 Fn (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ∈ V) → 𝐽 ∈ V)
31	6, 29, 30	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
32	31	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
33		setsabs 16925	. . . . . 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⟩))
34	28, 32, 33	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⟩))
35		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆)
36		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐶) = (Base‘𝐶)
37	35, 36	ressbas 16992	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑊 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑆)))
38	3, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶 ↾s 𝑆)))
39	38	sseq1d 3957	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇 ↔ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇))
40	39	biimpar 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇)
41		inss2 4169	. . . . . . . . . . 11 ⊢ (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
42	41	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
43	40, 42	ssind 4172	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (𝑇 ∩ (Base‘𝐶)))
44	4	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ⊆ 𝑆)
45	44	ssrind 4175	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶)))
46	43, 45	eqssd 3943	. . . . . . . 8 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) = (𝑇 ∩ (Base‘𝐶)))
47	46	oveq2d 7323	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶))))
48	3	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑆 ∈ 𝑊)
49	36	ressinbas 17000	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑊 → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
50	48, 49	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) = (𝐶 ↾s (𝑆 ∩ (Base‘𝐶))))
51	36	ressinbas 17000	. . . . . . . 8 ⊢ (𝑇 ∈ V → (𝐶 ↾s 𝑇) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶))))
52	10, 51	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑇) = (𝐶 ↾s (𝑇 ∩ (Base‘𝐶))))
53	47, 50, 52	3eqtr4d 2786	. . . . . 6 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑇))
54	53	oveq1d 7322	. . . . 5 ⊢ ((𝜑 ∧ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
55	27, 34, 54	3eqtrd 2780	. . . 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⟩))
56		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇)
57		ovexd 7342	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
58	5	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
59	11, 24	ressval2 16991	. . . . . . . 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)))⟩))
60	56, 57, 58, 59	syl3anc 1371	. . . . . . 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)))⟩))
61		ovexd 7342	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝐶 ↾s 𝑆) ∈ V)
62	23	necomi 2996	. . . . . . . . 9 ⊢ (Hom ‘ndx) ≠ (Base‘ndx)
63	62	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (Hom ‘ndx) ≠ (Base‘ndx))
64		rescabs.h	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
65	3, 3	xpexd 7633	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
66		fnex 7125	. . . . . . . . . 10 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
67	64, 65, 66	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ V)
68	67	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐻 ∈ V)
69		fvex 6817	. . . . . . . . . 10 ⊢ (Base‘(𝐶 ↾s 𝑆)) ∈ V
70	69	inex2 5251	. . . . . . . . 9 ⊢ (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆))) ∈ V
71	70	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆))) ∈ V)
72		fvex 6817	. . . . . . . . 9 ⊢ (Hom ‘ndx) ∈ V
73		fvex 6817	. . . . . . . . 9 ⊢ (Base‘ndx) ∈ V
74	72, 73	setscom 16926	. . . . . . . 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⟩))
75	61, 63, 68, 71, 74	syl22anc 837	. . . . . . 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⟩))
76		eqid 2736	. . . . . . . . . . 11 ⊢ ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) ↾s 𝑇)
77		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘(𝐶 ↾s 𝑆))
78	76, 77	ressval2 16991	. . . . . . . . . 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)))⟩))
79	56, 61, 58, 78	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = ((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩))
80	3	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑆 ∈ 𝑊)
81	4	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝑇 ⊆ 𝑆)
82		ressabs 17004	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑇 ⊆ 𝑆) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇))
83	80, 81, 82	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇))
84	79, 83	eqtr3d 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → ((𝐶 ↾s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶 ↾s 𝑆)))⟩) = (𝐶 ↾s 𝑇))
85	84	oveq1d 7322	. . . . . . 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⟩))
86	60, 75, 85	3eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩))
87	86	oveq1d 7322	. . . . 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⟩))
88		ovex 7340	. . . . . 6 ⊢ (𝐶 ↾s 𝑇) ∈ V
89	31	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Base‘(𝐶 ↾s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
90		setsabs 16925	. . . . . 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⟩))
91	88, 89, 90	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⟩))
92	87, 91	eqtrd 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⟩))
93	55, 92	pm2.61dan 811	. . 3 ⊢ (𝜑 → ((((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
94	7, 93	eqtrd 2776	. 2 ⊢ (𝜑 → (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
95		eqid 2736	. . . 4 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻)
96		rescabs.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
97	95, 96, 3, 64	rescval2 17585	. . 3 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
98	97	oveq1d 7322	. 2 ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽))
99		eqid 2736	. . 3 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽)
100	99, 96, 5, 6	rescval2 17585	. 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
101	94, 98, 100	3eqtr4d 2786	1 ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104   ≠ wne 2941  Vcvv 3437   ∩ cin 3891   ⊆ wss 3892  ⟨cop 4571   × cxp 5598   Fn wfn 6453  ‘cfv 6458  (class class class)co 7307  1c1 10918  4c4 12076  ;cdc 12483   sSet csts 16909  ndxcnx 16939  Basecbs 16957   ↾_s cress 16986  Hom chom 17018   ↾_cat cresc 17565

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-er 8529  df-en 8765  df-dom 8766  df-sdom 8767  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-nn 12020  df-2 12082  df-3 12083  df-4 12084  df-5 12085  df-6 12086  df-7 12087  df-8 12088  df-9 12089  df-n0 12280  df-z 12366  df-dec 12484  df-sets 16910  df-slot 16928  df-ndx 16940  df-base 16958  df-ress 16987  df-hom 17031  df-resc 17568

This theorem is referenced by: (None)