Theorem nllyrest 21099

Description: An open subspace of an n-locally 𝐴 space is also n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllyrest	⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ 𝑛-Locally 𝐴)

Proof of Theorem nllyrest

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

Step	Hyp	Ref	Expression
1		nllytop 21086	. . 3 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)
2		resttop 20774	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
3	1, 2	sylan 487	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
4		restopn2 20791	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
5	1, 4	sylan 487	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
6		simp1l 1078	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ 𝑛-Locally 𝐴)
7		simp2l 1080	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐽)
8		simp3 1056	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9		nlly2i 21089	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑠 ∈ 𝒫 𝑥∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))
10	6, 7, 8, 9	syl3anc 1318	. . . . . . . 8 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ∃𝑠 ∈ 𝒫 𝑥∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))
11	3	3ad2ant1 1075	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → (𝐽 ↾t 𝐵) ∈ Top)
12	11	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → (𝐽 ↾t 𝐵) ∈ Top)
13		simp3l 1082	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑢 ∈ 𝐽)
14		simp3r2 1163	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑢 ⊆ 𝑠)
15		simp2 1055	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑥)
16	15	elpwid 4118	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝑥)
17		simp12r 1168	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ⊆ 𝐵)
18	16, 17	sstrd 3578	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝐵)
19	14, 18	sstrd 3578	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑢 ⊆ 𝐵)
20	6	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝐽 ∈ 𝑛-Locally 𝐴)
21	20, 1	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝐽 ∈ Top)
22		simp11r 1166	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝐵 ∈ 𝐽)
23		restopn2 20791	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑢 ∈ (𝐽 ↾t 𝐵) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝐵)))
24	21, 22, 23	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → (𝑢 ∈ (𝐽 ↾t 𝐵) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝐵)))
25	13, 19, 24	mpbir2and 959	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑢 ∈ (𝐽 ↾t 𝐵))
26		simp3r1 1162	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑦 ∈ 𝑢)
27		opnneip 20733	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ↾t 𝐵) ∈ Top ∧ 𝑢 ∈ (𝐽 ↾t 𝐵) ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘(𝐽 ↾t 𝐵))‘{𝑦}))
28	12, 25, 26, 27	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑢 ∈ ((nei‘(𝐽 ↾t 𝐵))‘{𝑦}))
29		elssuni 4403	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ ∪ 𝐽)
30	22, 29	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝐵 ⊆ ∪ 𝐽)
31		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
32	31	restuni 20776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ ∪ 𝐽) → 𝐵 = ∪ (𝐽 ↾t 𝐵))
33	21, 30, 32	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝐵 = ∪ (𝐽 ↾t 𝐵))
34	18, 33	sseqtrd 3604	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ ∪ (𝐽 ↾t 𝐵))
35		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐽 ↾t 𝐵) = ∪ (𝐽 ↾t 𝐵)
36	35	ssnei2 20730	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ↾t 𝐵) ∈ Top ∧ 𝑢 ∈ ((nei‘(𝐽 ↾t 𝐵))‘{𝑦})) ∧ (𝑢 ⊆ 𝑠 ∧ 𝑠 ⊆ ∪ (𝐽 ↾t 𝐵))) → 𝑠 ∈ ((nei‘(𝐽 ↾t 𝐵))‘{𝑦}))
37	12, 28, 14, 34, 36	syl22anc 1319	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ∈ ((nei‘(𝐽 ↾t 𝐵))‘{𝑦}))
38	37, 15	elind 3760	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥))
39		restabs 20779	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ((𝐽 ↾t 𝐵) ↾t 𝑠) = (𝐽 ↾t 𝑠))
40	21, 18, 22, 39	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → ((𝐽 ↾t 𝐵) ↾t 𝑠) = (𝐽 ↾t 𝑠))
41		simp3r3 1164	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → (𝐽 ↾t 𝑠) ∈ 𝐴)
42	40, 41	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → ((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴)
43	38, 42	jca 553	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥 ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → (𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴))
44	43	3expa 1257	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴))) → (𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴))
45	44	rexlimdvaa 3014	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ 𝑠 ∈ 𝒫 𝑥) → (∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴) → (𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴)))
46	45	expimpd 627	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ((𝑠 ∈ 𝒫 𝑥 ∧ ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴)) → (𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴)))
47	46	reximdv2 2997	. . . . . . . 8 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → (∃𝑠 ∈ 𝒫 𝑥∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴) → ∃𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴))
48	10, 47	mpd 15	. . . . . . 7 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ∃𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴)
49	48	3expa 1257	. . . . . 6 ⊢ ((((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)) ∧ 𝑦 ∈ 𝑥) → ∃𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴)
50	49	ralrimiva 2949	. . . . 5 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)) → ∀𝑦 ∈ 𝑥 ∃𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴)
51	50	ex 449	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → ((𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴))
52	5, 51	sylbid 229	. . 3 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴))
53	52	ralrimiv 2948	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → ∀𝑥 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑥 ∃𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴)
54		isnlly 21082	. 2 ⊢ ((𝐽 ↾t 𝐵) ∈ 𝑛-Locally 𝐴 ↔ ((𝐽 ↾t 𝐵) ∈ Top ∧ ∀𝑥 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑥 ∃𝑠 ∈ (((nei‘(𝐽 ↾t 𝐵))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽 ↾t 𝐵) ↾t 𝑠) ∈ 𝐴))
55	3, 53, 54	sylanbrc 695	1 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ 𝑛-Locally 𝐴)

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  neicnei 20711  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-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-nei 20712  df-nlly 21080

This theorem is referenced by:  loclly  21100  nllyidm  21102