Theorem cldllycmp 21298

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 21289.) (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 21276	. . 3 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
2		resttop 20964	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ Top)
3	1, 2	sylan 488	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ Top)
4		elrest 16088	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝐴)))
5		simpll 790	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝐽 ∈ 𝑛-Locally Comp)
6		simprl 794	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑢 ∈ 𝐽)
7		inss1 3833	. . . . . . . . . . 11 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝑢
8		simprr 796	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑦 ∈ (𝑢 ∩ 𝐴))
9	7, 8	sseldi 3601	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑦 ∈ (𝑢 ∩ 𝐴))) → 𝑦 ∈ 𝑢)
10		nlly2i 21279	. . . . . . . . . 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 1326	. . . . . . . . 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 762	. . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . 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 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))) → 𝐴 ∈ (Clsd‘𝐽))
15		simprlr 803	. . . . . . . . . . . . . . . 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 16089	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽) ∧ 𝑤 ∈ 𝐽) → (𝑤 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
17	13, 14, 15, 16	syl3anc 1326	. . . . . . . . . . . . . . 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 1109	. . . . . . . . . . . . . . . 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 3834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∩ 𝐴) ⊆ 𝐴
20		simplrr 801	. . . . . . . . . . . . . . . . 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 3601	. . . . . . . . . . . . . . . 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 3798	. . . . . . . . . . . . . . 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 20923	. . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . 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 1110	. . . . . . . . . . . . . . 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 3838	. . . . . . . . . . . . . . 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 3834	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) ⊆ 𝐴
29		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
30	29	cldss 20833	. . . . . . . . . . . . . . . . 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 20966	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
33	13, 31, 32	syl2anc 693	. . . . . . . . . . . . . . 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 3653	. . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
36	35	ssnei2 20920	. . . . . . . . . . . . . 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 1327	. . . . . . . . . . . . 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 802	. . . . . . . . . . . . . . . 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 4170	. . . . . . . . . . . . . . 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 3838	. . . . . . . . . . . . . . 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 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑠 ∈ V
43	42	inex1 4799	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) ∈ V
44	43	elpw 4164	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∩ 𝐴) ∈ 𝒫 (𝑢 ∩ 𝐴) ↔ (𝑠 ∩ 𝐴) ⊆ (𝑢 ∩ 𝐴))
45	41, 44	sylibr 224	. . . . . . . . . . . . 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 3798	. . . . . . . . . . . 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 20969	. . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . 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 3833	. . . . . . . . . . . . . . . 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 20969	. . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . 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 2659	. . . . . . . . . . . . 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 1111	. . . . . . . . . . . . . 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 3805	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ 𝐴) = (𝐴 ∩ 𝑠)
57		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)
58		ineq1 3807	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → (𝑣 ∩ 𝑠) = (𝐴 ∩ 𝑠))
59	58	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → ((𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠) ↔ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)))
60	59	rspcev 3309	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∩ 𝑠) = (𝐴 ∩ 𝑠)) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠))
61	14, 57, 60	sylancl 694	. . . . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . . . . . 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 4467	. . . . . . . . . . . . . . . . . . 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 3613	. . . . . . . . . . . . . . . . 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 20976	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽) → ((𝐴 ∩ 𝑠) ∈ (Clsd‘(𝐽 ↾t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠)))
67	13, 65, 66	syl2anc 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))) → ((𝐴 ∩ 𝑠) ∈ (Clsd‘(𝐽 ↾t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴 ∩ 𝑠) = (𝑣 ∩ 𝑠)))
68	61, 67	mpbird 247	. . . . . . . . . . . . . . 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 2705	. . . . . . . . . . . . . 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 21205	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ↾t 𝑠) ∈ Comp ∧ (𝑠 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝑠))) → ((𝐽 ↾t 𝑠) ↾t (𝑠 ∩ 𝐴)) ∈ Comp)
71	55, 69, 70	syl2anc 693	. . . . . . . . . . . . 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 2701	. . . . . . . . . . . 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 6658	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑠 ∩ 𝐴) → ((𝐽 ↾t 𝐴) ↾t 𝑣) = ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)))
74	73	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑠 ∩ 𝐴) → (((𝐽 ↾t 𝐴) ↾t 𝑣) ∈ Comp ↔ ((𝐽 ↾t 𝐴) ↾t (𝑠 ∩ 𝐴)) ∈ Comp))
75	74	rspcev 3309	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 643	. . . . . . . . . 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 3038	. . . . . . . . 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 680	. . . . . . 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 2966	. . . . . 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 4161	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∩ 𝐴) → 𝒫 𝑥 = 𝒫 (𝑢 ∩ 𝐴))
83	82	ineq2d 3814	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ∩ 𝐴) → (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘(𝐽 ↾t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢 ∩ 𝐴)))
84	83	rexeqdv 3145	. . . . . . 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 3146	. . . . . 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 237	. . . . 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 3031	. . . 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 230	. . 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 2965	. 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 21272	. 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 698	1 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ 𝑛-Locally Comp)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ‘cfv 5888  (class class class)co 6650   ↾_t crest 16081  Topctop 20698  Clsdccld 20820  neicnei 20901  Compccmp 21189  𝑛-Locally cnlly 21268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-nei 20902  df-cmp 21190  df-nlly 21270

This theorem is referenced by:  rellycmp  22756