Theorem cldllycmp 21108

Description: A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 21099.) (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
cldllycmp	⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ 𝑛-Locally Comp)

Proof of Theorem cldllycmp

Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nllytop 21086	. . 3 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
2		resttop 20774	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ Top)
3	1, 2	sylan 487	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ Top)
4		elrest 15911	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝐴)))
5		simpll 786	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝐽 ∈ 𝑛-Locally Comp)
6		simprl 790	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑢 ∈ 𝐽)
7		inss1 3795	. . . . . . . . . . 11 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝑢
8		simprr 792	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑦 ∈ (𝑢 ∩ 𝐴))
9	7, 8	sseldi 3566	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑦 ∈ 𝑢)
10		nlly2i 21089	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑢 ∈ 𝐽 ∧ 𝑦 ∈ 𝑢) → ∃𝑠 ∈ 𝒫 𝑢∃𝑤 ∈ 𝐽 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))
11	5, 6, 9, 10	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → ∃𝑠 ∈ 𝒫 𝑢∃𝑤 ∈ 𝐽 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))
12	3	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝐽 ↾t 𝐴) ∈ Top)
13	1	ad3antrrr 762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝐽 ∈ Top)
14		simpllr 795	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝐴 ∈ (Clsd‘𝐽))
15		simprlr 799	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑤 ∈ 𝐽)
16		elrestr 15912	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽) ∧ 𝑤 ∈ 𝐽) → (𝑤 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
17	13, 14, 15, 16	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑤 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
18		simprr1 1102	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑦 ∈ 𝑤)
19		inss2 3796	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝐴
20		simplrr 797	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑢 ∩ 𝐴))
21	19, 20	sseldi 3566	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑦 ∈ 𝐴)
22	18, 21	elind 3760	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑤 ∩ 𝐴))
23		opnneip 20733	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ↾t 𝐴) ∈ Top ∧ (𝑤 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ 𝑦 ∈ (𝑤 ∩ 𝐴)) → (𝑤 ∩ 𝐴) ∈ ((nei‘(𝐽 ↾t 𝐴))‘{𝑦}))
24	12, 17, 22, 23	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑤 ∩ 𝐴) ∈ ((nei‘(𝐽 ↾t 𝐴))‘{𝑦}))
25		simprr2 1103	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑤 ⊆ 𝑠)
26		ssrin 3800	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ⊆ 𝑠 → (𝑤 ∩ 𝐴) ⊆ (𝑠 ∩ 𝐴))
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑤 ∩ 𝐴) ⊆ (𝑠 ∩ 𝐴))
28		inss2 3796	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) ⊆ 𝐴
29		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
30	29	cldss 20643	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
31	14, 30	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝐴 ⊆ ∪ 𝐽)
32	29	restuni 20776	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
33	13, 31, 32	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
34	28, 33	syl5sseq 3616	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑠 ∩ 𝐴) ⊆ ∪ (𝐽 ↾t 𝐴))
35		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
36	35	ssnei2 20730	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ↾t 𝐴) ∈ Top ∧ (𝑤 ∩ 𝐴) ∈ ((nei‘(𝐽 ↾t 𝐴))‘{𝑦})) ∧ ((𝑤 ∩ 𝐴) ⊆ (𝑠 ∩ 𝐴) ∧ (𝑠 ∩ 𝐴) ⊆ ∪ (𝐽 ↾t 𝐴))) → (𝑠 ∩ 𝐴) ∈ ((nei‘(𝐽 ↾t 𝐴))‘{𝑦}))
37	12, 24, 27, 34, 36	syl22anc 1319	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑠 ∩ 𝐴) ∈ ((nei‘(𝐽 ↾t 𝐴))‘{𝑦}))
38		simprll 798	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑠 ∈ 𝒫 𝑢)
39	38	elpwid 4118	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑠 ⊆ 𝑢)
40		ssrin 3800	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ⊆ 𝑢 → (𝑠 ∩ 𝐴) ⊆ (𝑢 ∩ 𝐴))
41	39, 40	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑠 ∩ 𝐴) ⊆ (𝑢 ∩ 𝐴))
42		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑠 ∈ V
43	42	inex1 4727	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) ∈ V
44	43	elpw 4114	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∩ 𝐴) ∈ 𝒫 (𝑢 ∩ 𝐴) ↔ (𝑠 ∩ 𝐴) ⊆ (𝑢 ∩ 𝐴))
45	41, 44	sylibr 223	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑠 ∩ 𝐴) ∈ 𝒫 (𝑢 ∩ 𝐴))
46	37, 45	elind 3760	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑠 ∩ 𝐴) ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴)))
47	28	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑠 ∩ 𝐴) ⊆ 𝐴)
48		restabs 20779	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ (𝑠 ∩ 𝐴) ⊆ 𝐴 ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) = (𝐽 ↾t (𝑠 ∩ 𝐴)))
49	13, 47, 14, 48	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) = (𝐽 ↾t (𝑠 ∩ 𝐴)))
50		inss1 3795	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∩ 𝐴) ⊆ 𝑠
51	50	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑠 ∩ 𝐴) ⊆ 𝑠)
52		restabs 20779	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ (𝑠 ∩ 𝐴) ⊆ 𝑠 ∧ 𝑠 ∈ 𝒫 𝑢) → ((𝐽 ↾t 𝑠) ↾t (𝑠 ∩ 𝐴)) = (𝐽 ↾t (𝑠 ∩ 𝐴)))
53	13, 51, 38, 52	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → ((𝐽 ↾t 𝑠) ↾t (𝑠 ∩ 𝐴)) = (𝐽 ↾t (𝑠 ∩ 𝐴)))
54	49, 53	eqtr4d 2647	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) = ((𝐽 ↾t 𝑠) ↾t (𝑠 ∩ 𝐴)))
55		simprr3 1104	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝐽 ↾t 𝑠) ∈ Comp)
56		incom 3767	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) = (𝐴 ∩ 𝑠)
57		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)
58		ineq1 3769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → (𝑣 ∩ 𝑠) = (𝐴 ∩ 𝑠))
59	58	eqeq2d 2620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → ((𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠) ↔ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)))
60	59	rspcev 3282	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠))
61	14, 57, 60	sylancl 693	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠))
62		simplrl 796	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑢 ∈ 𝐽)
63		elssuni 4403	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝐽 → 𝑢 ⊆ ∪ 𝐽)
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑢 ⊆ ∪ 𝐽)
65	39, 64	sstrd 3578	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → 𝑠 ⊆ ∪ 𝐽)
66	29	restcld 20786	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽) → ((𝐴 ∩ 𝑠) ∈ (Clsd‘(𝐽 ↾t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠)))
67	13, 65, 66	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → ((𝐴 ∩ 𝑠) ∈ (Clsd‘(𝐽 ↾t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠)))
68	61, 67	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝐴 ∩ 𝑠) ∈ (Clsd‘(𝐽 ↾t 𝑠)))
69	56, 68	syl5eqel 2692	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → (𝑠 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝑠)))
70		cmpcld 21015	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ↾t 𝑠) ∈ Comp ∧ (𝑠 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝑠))) → ((𝐽 ↾t 𝑠) ↾t (𝑠 ∩ 𝐴)) ∈ Comp)
71	55, 69, 70	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → ((𝐽 ↾t 𝑠) ↾t (𝑠 ∩ 𝐴)) ∈ Comp)
72	54, 71	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) ∈ Comp)
73		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑠 ∩ 𝐴) → ((𝐽 ↾t 𝐴) ↾t 𝑣) = ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)))
74	73	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑠 ∩ 𝐴) → (((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp ↔ ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) ∈ Comp))
75	74	rspcev 3282	. . . . . . . . . . . 12 ⊢ (((𝑠 ∩ 𝐴) ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴)) ∧ ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) ∈ Comp) → ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp)
76	46, 72, 75	syl2anc 691	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))) → ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp)
77	76	expr 641	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) ∧ (𝑠 ∈ 𝒫 𝑢 ∧ 𝑤 ∈ 𝐽)) → ((𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp) → ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp))
78	77	rexlimdvva 3020	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → (∃𝑠 ∈ 𝒫 𝑢∃𝑤 ∈ 𝐽 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp) → ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp))
79	11, 78	mpd 15	. . . . . . . 8 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp)
80	79	anassrs 678	. . . . . . 7 ⊢ ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝑢 ∩ 𝐴)) → ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp)
81	80	ralrimiva 2949	. . . . . 6 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ (𝑢 ∩ 𝐴)∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp)
82		pweq 4111	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∩ 𝐴) → 𝒫 𝑥 = 𝒫 (𝑢 ∩ 𝐴))
83	82	ineq2d 3776	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ∩ 𝐴) → (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴)))
84	83	rexeqdv 3122	. . . . . . 7 ⊢ (𝑥 = (𝑢 ∩ 𝐴) → (∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp ↔ ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp))
85	84	raleqbi1dv 3123	. . . . . 6 ⊢ (𝑥 = (𝑢 ∩ 𝐴) → (∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp ↔ ∀𝑦 ∈ (𝑢 ∩ 𝐴)∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴))((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp))
86	81, 85	syl5ibrcom 236	. . . . 5 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢 ∈ 𝐽) → (𝑥 = (𝑢 ∩ 𝐴) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp))
87	86	rexlimdva 3013	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝐴) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp))
88	4, 87	sylbid 229	. . 3 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽 ↾t 𝐴) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp))
89	88	ralrimiv 2948	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∀𝑥 ∈ (𝐽 ↾t 𝐴)∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp)
90		isnlly 21082	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ 𝑛-Locally Comp ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ ∀𝑥 ∈ (𝐽 ↾t 𝐴)∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp))
91	3, 89, 90	sylanbrc 695	1 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ 𝑛-Locally Comp)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  Topctop 20517  Clsdccld 20630  neicnei 20711  Compccmp 20999  𝑛-Locally cnlly 21078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-nei 20712  df-cmp 21000  df-nlly 21080

This theorem is referenced by:  rellycmp  22564