Theorem nllyrest 23451

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 23438	. . 3 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)
2		resttop 23125	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
3	1, 2	sylan 581	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
4		restopn2 23142	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
5	1, 4	sylan 581	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
6		simp1l 1199	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ 𝑛-Locally 𝐴)
7		simp2l 1201	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐽)
8		simp3 1139	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9		nlly2i 23441	. . . . . . . . 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 1374	. . . . . . . 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 1134	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → (𝐽 ↾t 𝐵) ∈ Top)
12	11	3ad2ant1 1134	. . . . . . . . . . . . . . 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 1203	. . . . . . . . . . . . . . . . 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 1284	. . . . . . . . . . . . . . . . . 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 1138	. . . . . . . . . . . . . . . . . . . 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 4550	. . . . . . . . . . . . . . . . . . 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 1289	. . . . . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . 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 1287	. . . . . . . . . . . . . . . . . 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 23142	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑢 ∈ (𝐽 ↾t 𝐵) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝐵)))
24	21, 22, 23	syl2anc 585	. . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . 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 1283	. . . . . . . . . . . . . . . 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 23084	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ↾t 𝐵) ∈ Top ∧ 𝑢 ∈ (𝐽 ↾t 𝐵) ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘(𝐽 ↾t 𝐵))‘{𝑦}))
28	12, 25, 26, 27	syl3anc 1374	. . . . . . . . . . . . . . 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 4881	. . . . . . . . . . . . . . . . . 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 23127	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ ∪ 𝐽) → 𝐵 = ∪ (𝐽 ↾t 𝐵))
33	21, 30, 32	syl2anc 585	. . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . 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 23081	. . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . 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 23130	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ((𝐽 ↾t 𝐵) ↾t 𝑠) = (𝐽 ↾t 𝑠))
40	21, 18, 22, 39	syl3anc 1374	. . . . . . . . . . . . . 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 1285	. . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 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 1119	. . . . . . . . . . 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 3139	. . . . . . . . . 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 3147	. . . . . . . 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 1119	. . . . . 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 3129	. . . . 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 3128	. 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 23434	. 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 584	1 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ 𝑛-Locally 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  {csn 4567  ∪ cuni 4850  ‘cfv 6498  (class class class)co 7367   ↾_t crest 17383  Topctop 22858  neicnei 23062  𝑛-Locally cnlly 23430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-en 8894  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-nei 23063  df-nlly 23432

This theorem is referenced by:  loclly  23452  nllyidm  23454