Theorem nllyrest 23429

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 23416	. . 3 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)
2		resttop 23103	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
3	1, 2	sylan 580	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
4		restopn2 23120	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
5	1, 4	sylan 580	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
6		simp1l 1198	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ 𝑛-Locally 𝐴)
7		simp2l 1200	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐽)
8		simp3 1138	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9		nlly2i 23419	. . . . . . . . 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 1373	. . . . . . . 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 1133	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → (𝐽 ↾t 𝐵) ∈ Top)
12	11	3ad2ant1 1133	. . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . . 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 1283	. . . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . . . . 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 4589	. . . . . . . . . . . . . . . . . . 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 1288	. . . . . . . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . 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 1286	. . . . . . . . . . . . . . . . . 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 23120	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑢 ∈ (𝐽 ↾t 𝐵) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝐵)))
24	21, 22, 23	syl2anc 584	. . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . 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 1282	. . . . . . . . . . . . . . . 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 23062	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ↾t 𝐵) ∈ Top ∧ 𝑢 ∈ (𝐽 ↾t 𝐵) ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘(𝐽 ↾t 𝐵))‘{𝑦}))
28	12, 25, 26, 27	syl3anc 1373	. . . . . . . . . . . . . . 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 4918	. . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
32	31	restuni 23105	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ ∪ 𝐽) → 𝐵 = ∪ (𝐽 ↾t 𝐵))
33	21, 30, 32	syl2anc 584	. . . . . . . . . . . . . . . 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 4000	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐽 ↾t 𝐵) = ∪ (𝐽 ↾t 𝐵)
36	35	ssnei2 23059	. . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 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 4180	. . . . . . . . . . . . 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 23108	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ((𝐽 ↾t 𝐵) ↾t 𝑠) = (𝐽 ↾t 𝑠))
40	21, 18, 22, 39	syl3anc 1373	. . . . . . . . . . . . . 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 1284	. . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 1118	. . . . . . . . . . 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 3143	. . . . . . . . . 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 453	. . . . . . . . 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 3151	. . . . . . . 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 1118	. . . . . 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 3133	. . . . 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 412	. . . 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 240	. . 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 3132	. 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 23412	. 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 583	1 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ 𝑛-Locally 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580  {csn 4606  ∪ cuni 4888  ‘cfv 6536  (class class class)co 7410   ↾_t crest 17439  Topctop 22836  neicnei 23040  𝑛-Locally cnlly 23408

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-en 8965  df-fin 8968  df-fi 9428  df-rest 17441  df-topgen 17462  df-top 22837  df-topon 22854  df-bases 22889  df-nei 23041  df-nlly 23410

This theorem is referenced by:  loclly  23430  nllyidm  23432