Theorem cldllycmp 23519

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 23510.) (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 23497	. . 3 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
2		resttop 23184	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ Top)
3	1, 2	sylan 580	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ Top)
4		elrest 17474	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝐴)))
5		simpll 767	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝐽 ∈ 𝑛-Locally Comp)
6		simprl 771	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑢 ∈ 𝐽)
7		simprr 773	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑦 ∈ (𝑢 ∩ 𝐴))
8	7	elin1d 4214	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑦 ∈ 𝑢)
9		nlly2i 23500	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑢 ∈ 𝐽 ∧ 𝑦 ∈ 𝑢) → ∃𝑠 ∈ 𝒫 𝑢∃𝑤 ∈ 𝐽 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ Comp))
10	5, 6, 8, 9	syl3anc 1370	. . . . . . . . 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))
11	3	ad2antrr 726	. . . . . . . . . . . . . 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)
12	1	ad3antrrr 730	. . . . . . . . . . . . . . . 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)
13		simpllr 776	. . . . . . . . . . . . . . . 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))
14		simprlr 780	. . . . . . . . . . . . . . . 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)
15		elrestr 17475	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽) ∧ 𝑤 ∈ 𝐽) → (𝑤 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
16	12, 13, 14, 15	syl3anc 1370	. . . . . . . . . . . . . . 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))
17		simprr1 1220	. . . . . . . . . . . . . . . 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)
18		simplrr 778	. . . . . . . . . . . . . . . . 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))
19	18	elin2d 4215	. . . . . . . . . . . . . . . 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)
20	17, 19	elind 4210	. . . . . . . . . . . . . . 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))
21		opnneip 23143	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ↾t 𝐴) ∈ Top ∧ (𝑤 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ 𝑦 ∈ (𝑤 ∩ 𝐴)) → (𝑤 ∩ 𝐴) ∈ ((nei‘(𝐽 ↾t 𝐴))‘{𝑦}))
22	11, 16, 20, 21	syl3anc 1370	. . . . . . . . . . . . . 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}))
23		simprr2 1221	. . . . . . . . . . . . . . 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)
24	23	ssrind 4252	. . . . . . . . . . . . . 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))
25		inss2 4246	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) ⊆ 𝐴
26		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
27	26	cldss 23053	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
28	13, 27	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)
29	26	restuni 23186	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
30	12, 28, 29	syl2anc 584	. . . . . . . . . . . . . . 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))
31	25, 30	sseqtrid 4048	. . . . . . . . . . . . . 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))
32		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
33	32	ssnei2 23140	. . . . . . . . . . . . . 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}))
34	11, 22, 24, 31, 33	syl22anc 839	. . . . . . . . . . . . 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}))
35		simprll 779	. . . . . . . . . . . . . . . 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)
36	35	elpwid 4614	. . . . . . . . . . . . . . 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)
37	36	ssrind 4252	. . . . . . . . . . . . . 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))
38		vex 3482	. . . . . . . . . . . . . . . 16 ⊢ 𝑠 ∈ V
39	38	inex1 5323	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) ∈ V
40	39	elpw 4609	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∩ 𝐴) ∈ 𝒫 (𝑢 ∩ 𝐴) ↔ (𝑠 ∩ 𝐴) ⊆ (𝑢 ∩ 𝐴))
41	37, 40	sylibr 234	. . . . . . . . . . . . 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))
42	34, 41	elind 4210	. . . . . . . . . . . 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)))
43	25	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)
44		restabs 23189	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ (𝑠 ∩ 𝐴) ⊆ 𝐴 ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) = (𝐽 ↾t (𝑠 ∩ 𝐴)))
45	12, 43, 13, 44	syl3anc 1370	. . . . . . . . . . . . . 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)))
46		inss1 4245	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∩ 𝐴) ⊆ 𝑠
47	46	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)
48		restabs 23189	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ (𝑠 ∩ 𝐴) ⊆ 𝑠 ∧ 𝑠 ∈ 𝒫 𝑢) → ((𝐽 ↾t 𝑠) ↾t (𝑠 ∩ 𝐴)) = (𝐽 ↾t (𝑠 ∩ 𝐴)))
49	12, 47, 35, 48	syl3anc 1370	. . . . . . . . . . . . . 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)))
50	45, 49	eqtr4d 2778	. . . . . . . . . . . . 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)))
51		simprr3 1222	. . . . . . . . . . . . . 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)
52		incom 4217	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) = (𝐴 ∩ 𝑠)
53		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)
54		ineq1 4221	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (𝑣 ∩ 𝑠) = (𝐴 ∩ 𝑠))
55	54	rspceeqv 3645	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠))
56	13, 53, 55	sylancl 586	. . . . . . . . . . . . . . . 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))
57		simplrl 777	. . . . . . . . . . . . . . . . . . 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)
58		elssuni 4942	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝐽 → 𝑢 ⊆ ∪ 𝐽)
59	57, 58	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)
60	36, 59	sstrd 4006	. . . . . . . . . . . . . . . . 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)
61	26	restcld 23196	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽) → ((𝐴 ∩ 𝑠) ∈ (Clsd‘(𝐽 ↾t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠)))
62	12, 60, 61	syl2anc 584	. . . . . . . . . . . . . . . 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)))
63	56, 62	mpbird 257	. . . . . . . . . . . . . . 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)))
64	52, 63	eqeltrid 2843	. . . . . . . . . . . . . 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)))
65		cmpcld 23426	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ↾t 𝑠) ∈ Comp ∧ (𝑠 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝑠))) → ((𝐽 ↾t 𝑠) ↾t (𝑠 ∩ 𝐴)) ∈ Comp)
66	51, 64, 65	syl2anc 584	. . . . . . . . . . . . 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)
67	50, 66	eqeltrd 2839	. . . . . . . . . . . 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)
68		oveq2 7439	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑠 ∩ 𝐴) → ((𝐽 ↾t 𝐴) ↾t 𝑣) = ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)))
69	68	eleq1d 2824	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑠 ∩ 𝐴) → (((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp ↔ ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) ∈ Comp))
70	69	rspcev 3622	. . . . . . . . . . . 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)
71	42, 67, 70	syl2anc 584	. . . . . . . . . . 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)
72	71	expr 456	. . . . . . . . . 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))
73	72	rexlimdvva 3211	. . . . . . . . 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))
74	10, 73	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)
75	74	anassrs 467	. . . . . . 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)
76	75	ralrimiva 3144	. . . . . 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)
77		pweq 4619	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∩ 𝐴) → 𝒫 𝑥 = 𝒫 (𝑢 ∩ 𝐴))
78	77	ineq2d 4228	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ∩ 𝐴) → (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴)))
79	78	rexeqdv 3325	. . . . . . 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))
80	79	raleqbi1dv 3336	. . . . . 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))
81	76, 80	syl5ibrcom 247	. . . . 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))
82	81	rexlimdva 3153	. . . 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))
83	4, 82	sylbid 240	. . 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))
84	83	ralrimiv 3143	. 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)
85		isnlly 23493	. 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))
86	3, 84, 85	sylanbrc 583	1 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ 𝑛-Locally Comp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912  ‘cfv 6563  (class class class)co 7431   ↾_t crest 17467  Topctop 22915  Clsdccld 23040  neicnei 23121  Compccmp 23410  𝑛-Locally cnlly 23489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-nei 23122  df-cmp 23411  df-nlly 23491

This theorem is referenced by:  rellycmp  25003