Theorem nllyrest 23811

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 23798	. . 3 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)
2		resttop 23484	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
3	1, 2	sylan 592	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
4		restopn2 23501	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
5	1, 4	sylan 592	. . . 4 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
6		simp1l 1216	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ 𝑛-Locally 𝐴)
7		simp2l 1218	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐽)
8		simp3 1156	. . . . . . . . 9 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9		nlly2i 23801	. . . . . . . . 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 1398	. . . . . . . 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 1151	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → (𝐽 ↾t 𝐵) ∈ Top)
12	11	3ad2ant1 1151	. . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . . 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 1301	. . . . . . . . . . . . . . . . . 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 1155	. . . . . . . . . . . . . . . . . . . 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 4565	. . . . . . . . . . . . . . . . . . 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 1306	. . . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 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 1304	. . . . . . . . . . . . . . . . . 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 23501	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑢 ∈ (𝐽 ↾t 𝐵) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝐵)))
24	21, 22, 23	syl2anc 596	. . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 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 1300	. . . . . . . . . . . . . . . 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 23443	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ↾t 𝐵) ∈ Top ∧ 𝑢 ∈ (𝐽 ↾t 𝐵) ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘(𝐽 ↾t 𝐵))‘{𝑦}))
28	12, 25, 26, 27	syl3anc 1398	. . . . . . . . . . . . . . 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 4898	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ ∪ 𝐽)
30	22, 29	syl 18	. . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 = ∪ 𝐽
32	31	restuni 23486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ ∪ 𝐽) → 𝐵 = ∪ (𝐽 ↾t 𝐵))
33	21, 30, 32	syl2anc 596	. . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐽 ↾t 𝐵) = ∪ (𝐽 ↾t 𝐵)
36	35	ssnei2 23440	. . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . 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 4097	. . . . . . . . . . . . 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 23489	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ((𝐽 ↾t 𝐵) ↾t 𝑠) = (𝐽 ↾t 𝑠))
40	21, 18, 22, 39	syl3anc 1398	. . . . . . . . . . . . . 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 1302	. . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . 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 521	. . . . . . . . . . . 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 1136	. . . . . . . . . . 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 3164	. . . . . . . . . 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 459	. . . . . . . . 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 3172	. . . . . . . 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 16	. . . . . . 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 1136	. . . . . 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 3154	. . . . 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 418	. . . 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 243	. . 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 3153	. 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 23794	. 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 595	1 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ 𝑛-Locally 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  {csn 4584  ∪ cuni 4866  ‘cfv 6539  (class class class)co 7421   ↾_t crest 17597  Topctop 23217  neicnei 23421  𝑛-Locally cnlly 23790

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-en 8978  df-fin 8981  df-fi 9407  df-rest 17599  df-topgen 17620  df-top 23218  df-topon 23235  df-bases 23270  df-nei 23422  df-nlly 23792

This theorem is used by:  loclly  23812  nllyidm  23814