Theorem nllyidm 22094

Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 22092 to show 𝑛-Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴.) (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllyidm	⊢ Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴

Proof of Theorem nllyidm

Dummy variables 𝑗 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		llytop 22077	. . . 4 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → 𝑗 ∈ Top)
2		llyi 22079	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))
3		simprr3 1220	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴)
4		simprl 770	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ∈ 𝑗)
5		ssidd 3938	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ⊆ 𝑢)
6		simpl1 1188	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Locally 𝑛-Locally 𝐴)
7	6, 1	syl 17	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Top)
8		restopn2 21782	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
9	7, 4, 8	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
10	4, 5, 9	mpbir2and 712	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ∈ (𝑗 ↾t 𝑢))
11		simprr2 1219	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑦 ∈ 𝑢)
12		nlly2i 22081	. . . . . . . . 9 ⊢ (((𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴 ∧ 𝑢 ∈ (𝑗 ↾t 𝑢) ∧ 𝑦 ∈ 𝑢) → ∃𝑣 ∈ 𝒫 𝑢∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))
13	3, 10, 11, 12	syl3anc 1368	. . . . . . . 8 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ 𝒫 𝑢∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))
14		restopn2 21782	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑧 ∈ (𝑗 ↾t 𝑢) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢)))
15	7, 4, 14	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑧 ∈ (𝑗 ↾t 𝑢) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢)))
16	15	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗 ↾t 𝑢) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢)))
17	7	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
18		simpr2l 1229	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧 ∈ 𝑗)
19		simpr31 1260	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦 ∈ 𝑧)
20		opnneip 21724	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ Top ∧ 𝑧 ∈ 𝑗 ∧ 𝑦 ∈ 𝑧) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
21	17, 18, 19, 20	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
22		simpr32 1261	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧 ⊆ 𝑣)
23		simpr1 1191	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑢)
24	23	elpwid 4508	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝑢)
25	4	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 ∈ 𝑗)
26		elssuni 4830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝑗 → 𝑢 ⊆ ∪ 𝑗)
27	25, 26	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 ⊆ ∪ 𝑗)
28	24, 27	sstrd 3925	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ ∪ 𝑗)
29		eqid 2798	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑗 = ∪ 𝑗
30	29	ssnei2 21721	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 ∈ Top ∧ 𝑧 ∈ ((nei‘𝑗)‘{𝑦})) ∧ (𝑧 ⊆ 𝑣 ∧ 𝑣 ⊆ ∪ 𝑗)) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
31	17, 21, 22, 28, 30	syl22anc 837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
32		simprr1 1218	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ⊆ 𝑥)
33	32	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 ⊆ 𝑥)
34	24, 33	sstrd 3925	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝑥)
35		velpw 4502	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥)
36	34, 35	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
37	31, 36	elind 4121	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥))
38		restabs 21770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ Top ∧ 𝑣 ⊆ 𝑢 ∧ 𝑢 ∈ 𝑗) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
39	17, 24, 25, 38	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
40		simpr33 1262	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴)
41	39, 40	eqeltrrd 2891	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗 ↾t 𝑣) ∈ 𝐴)
42	37, 41	jca 515	. . . . . . . . . . . . . 14 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))
43	42	3exp2 1351	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑣 ∈ 𝒫 𝑢 → ((𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))))
44	43	imp 410	. . . . . . . . . . . 12 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → ((𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))))
45	16, 44	sylbid 243	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗 ↾t 𝑢) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴))))
46	45	rexlimdv 3242	. . . . . . . . . 10 ⊢ ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))
47	46	expimpd 457	. . . . . . . . 9 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → ((𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗 ↾t 𝑣) ∈ 𝐴)))
48	47	reximdv2 3230	. . . . . . . 8 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → (∃𝑣 ∈ 𝒫 𝑢∃𝑧 ∈ (𝑗 ↾t 𝑢)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑣 ∧ ((𝑗 ↾t 𝑢) ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴))
49	13, 48	mpd 15	. . . . . . 7 ⊢ (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝑗 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
50	2, 49	rexlimddv 3250	. . . . . 6 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
51	50	3expb 1117	. . . . 5 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
52	51	ralrimivva 3156	. . . 4 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
53		isnlly 22074	. . . 4 ⊢ (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴))
54	1, 52, 53	sylanbrc 586	. . 3 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → 𝑗 ∈ 𝑛-Locally 𝐴)
55	54	ssriv 3919	. 2 ⊢ Locally 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐴
56		nllyrest 22091	. . . . 5 ⊢ ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
57	56	adantl 485	. . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
58		nllytop 22078	. . . . . 6 ⊢ (𝑗 ∈ 𝑛-Locally 𝐴 → 𝑗 ∈ Top)
59	58	ssriv 3919	. . . . 5 ⊢ 𝑛-Locally 𝐴 ⊆ Top
60	59	a1i 11	. . . 4 ⊢ (⊤ → 𝑛-Locally 𝐴 ⊆ Top)
61	57, 60	restlly 22088	. . 3 ⊢ (⊤ → 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴)
62	61	mptru 1545	. 2 ⊢ 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴
63	55, 62	eqssi 3931	1 ⊢ Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800  ‘cfv 6324  (class class class)co 7135   ↾_t crest 16686  Topctop 21498  neicnei 21702  Locally clly 22069  𝑛-Locally cnlly 22070

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-en 8493  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-nei 21703  df-lly 22071  df-nlly 22072

This theorem is referenced by: (None)