Theorem nllyidm 23863

Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 23861 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 23846	. . . 4 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → 𝑗 ∈ Top)
2		llyi 23848	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝑛-Locally 𝐴))
3		simprr3 1242	. . . . . . . . 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 783	. . . . . . . . . 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 3936	. . . . . . . . . 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 1210	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 23550	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑢 ∈ (𝑗 ↾t 𝑢) ↔ (𝑢 ∈ 𝑗 ∧ 𝑢 ⊆ 𝑢)))
9	7, 4, 8	syl2anc 596	. . . . . . . . . 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 726	. . . . . . . . 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 1241	. . . . . . . . 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 23850	. . . . . . . . 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 1399	. . . . . . . 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 23550	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑧 ∈ (𝑗 ↾t 𝑢) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ⊆ 𝑢)))
15	7, 4, 14	syl2anc 596	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 1251	. . . . . . . . . . . . . . . . . 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 1282	. . . . . . . . . . . . . . . . . 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 23492	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ Top ∧ 𝑧 ∈ 𝑗 ∧ 𝑦 ∈ 𝑧) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
21	17, 18, 19, 20	syl3anc 1399	. . . . . . . . . . . . . . . . 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 1283	. . . . . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . . . . . . . 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 4569	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . 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 4904	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝑗 → 𝑢 ⊆ ∪ 𝑗)
27	25, 26	syl 18	. . . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑗 = ∪ 𝑗
30	29	ssnei2 23489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 ∈ Top ∧ 𝑧 ∈ ((nei‘𝑗)‘{𝑦})) ∧ (𝑧 ⊆ 𝑣 ∧ 𝑣 ⊆ ∪ 𝑗)) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
31	17, 21, 22, 28, 30	syl22anc 852	. . . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . . 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 4565	. . . . . . . . . . . . . . . . 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 4100	. . . . . . . . . . . . . . 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 23538	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ Top ∧ 𝑣 ⊆ 𝑢 ∧ 𝑢 ∈ 𝑗) → ((𝑗 ↾t 𝑢) ↾t 𝑣) = (𝑗 ↾t 𝑣))
39	17, 24, 25, 38	syl3anc 1399	. . . . . . . . . . . . . . . 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 1284	. . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 3165	. . . . . . . . . 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 459	. . . . . . . . 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 3176	. . . . . . . 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 16	. . . . . . 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 3173	. . . . . 6 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
51	50	3expb 1138	. . . . 5 ⊢ ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
52	51	ralrimivva 3209	. . . 4 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑣) ∈ 𝐴)
53		isnlly 23843	. . . 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 595	. . 3 ⊢ (𝑗 ∈ Locally 𝑛-Locally 𝐴 → 𝑗 ∈ 𝑛-Locally 𝐴)
55	54	ssriv 3917	. 2 ⊢ Locally 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐴
56		nllyrest 23860	. . . . 5 ⊢ ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
57	56	adantl 487	. . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ 𝑛-Locally 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝑛-Locally 𝐴)
58		nllytop 23847	. . . . . 6 ⊢ (𝑗 ∈ 𝑛-Locally 𝐴 → 𝑗 ∈ Top)
59	58	ssriv 3917	. . . . 5 ⊢ 𝑛-Locally 𝐴 ⊆ Top
60	59	a1i 11	. . . 4 ⊢ (⊤ → 𝑛-Locally 𝐴 ⊆ Top)
61	57, 60	restlly 23857	. . 3 ⊢ (⊤ → 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴)
62	61	mptru 1577	. 2 ⊢ 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴
63	55, 62	eqssi 3929	1 ⊢ Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  {csn 4588  ∪ cuni 4872  ‘cfv 6566  (class class class)co 7451   ↾_t crest 17641  Topctop 23266  neicnei 23470  Locally clly 23838  𝑛-Locally cnlly 23839

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-en 9012  df-fin 9015  df-fi 9440  df-rest 17643  df-topgen 17664  df-top 23267  df-topon 23284  df-bases 23319  df-nei 23471  df-lly 23840  df-nlly 23841

This theorem is used by: (None)