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Theorem nllyidm 21465
Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 21463 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.
StepHypRef Expression
1 llytop 21448 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴𝑗 ∈ Top)
2 llyi 21450 . . . . . . 7 ((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑢𝑗 (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))
3 simprr3 1253 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴)
4 simprl 811 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑗)
5 ssid 3753 . . . . . . . . . . 11 𝑢𝑢
65a1i 11 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑢)
7 simpl1 1204 . . . . . . . . . . . 12 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Locally 𝑛-Locally 𝐴)
87, 1syl 17 . . . . . . . . . . 11 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Top)
9 restopn2 21154 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
108, 4, 9syl2anc 696 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
114, 6, 10mpbir2and 995 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ∈ (𝑗t 𝑢))
12 simprr2 1251 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑦𝑢)
13 nlly2i 21452 . . . . . . . . 9 (((𝑗t 𝑢) ∈ 𝑛-Locally 𝐴𝑢 ∈ (𝑗t 𝑢) ∧ 𝑦𝑢) → ∃𝑣 ∈ 𝒫 𝑢𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
143, 11, 12, 13syl3anc 1463 . . . . . . . 8 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ 𝒫 𝑢𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
15 restopn2 21154 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
168, 4, 15syl2anc 696 . . . . . . . . . . . . 13 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
1716adantr 472 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
188adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
19 simpr2l 1271 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧𝑗)
20 simpr31 1321 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦𝑧)
21 opnneip 21096 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑧𝑗𝑦𝑧) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
2218, 19, 20, 21syl3anc 1463 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
23 simpr32 1323 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧𝑣)
24 simpr1 1210 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑢)
2524elpwid 4302 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑢)
264adantr 472 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑗)
27 elssuni 4607 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑗𝑢 𝑗)
2826, 27syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 𝑗)
2925, 28sstrd 3742 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 𝑗)
30 eqid 2748 . . . . . . . . . . . . . . . . . 18 𝑗 = 𝑗
3130ssnei2 21093 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ Top ∧ 𝑧 ∈ ((nei‘𝑗)‘{𝑦})) ∧ (𝑧𝑣𝑣 𝑗)) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
3218, 22, 23, 29, 31syl22anc 1464 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
33 simprr1 1249 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑥)
3433adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑥)
3525, 34sstrd 3742 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
36 selpw 4297 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
3735, 36sylibr 224 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
3832, 37elind 3929 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥))
39 restabs 21142 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ Top ∧ 𝑣𝑢𝑢𝑗) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
4018, 25, 26, 39syl3anc 1463 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
41 simpr33 1325 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)
4240, 41eqeltrrd 2828 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗t 𝑣) ∈ 𝐴)
4338, 42jca 555 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴))
44433exp2 1433 . . . . . . . . . . . . 13 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑣 ∈ 𝒫 𝑢 → ((𝑧𝑗𝑧𝑢) → ((𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴)))))
4544imp 444 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → ((𝑧𝑗𝑧𝑢) → ((𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴))))
4617, 45sylbid 230 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗t 𝑢) → ((𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴))))
4746rexlimdv 3156 . . . . . . . . . 10 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (∃𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴)))
4847expimpd 630 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → ((𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴)))
4948reximdv2 3140 . . . . . . . 8 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (∃𝑣 ∈ 𝒫 𝑢𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴))
5014, 49mpd 15 . . . . . . 7 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
512, 50rexlimddv 3161 . . . . . 6 ((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
52513expb 1113 . . . . 5 ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
5352ralrimivva 3097 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴 → ∀𝑥𝑗𝑦𝑥𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
54 isnlly 21445 . . . 4 (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴))
551, 53, 54sylanbrc 701 . . 3 (𝑗 ∈ Locally 𝑛-Locally 𝐴𝑗 ∈ 𝑛-Locally 𝐴)
5655ssriv 3736 . 2 Locally 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐴
57 nllyrest 21462 . . . . 5 ((𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
5857adantl 473 . . . 4 ((⊤ ∧ (𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
59 nllytop 21449 . . . . . 6 (𝑗 ∈ 𝑛-Locally 𝐴𝑗 ∈ Top)
6059ssriv 3736 . . . . 5 𝑛-Locally 𝐴 ⊆ Top
6160a1i 11 . . . 4 (⊤ → 𝑛-Locally 𝐴 ⊆ Top)
6258, 61restlly 21459 . . 3 (⊤ → 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴)
6362trud 1630 . 2 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴
6456, 63eqssi 3748 1 Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wtru 1621  wcel 2127  wral 3038  wrex 3039  cin 3702  wss 3703  𝒫 cpw 4290  {csn 4309   cuni 4576  cfv 6037  (class class class)co 6801  t crest 16254  Topctop 20871  neicnei 21074  Locally clly 21440  𝑛-Locally cnlly 21441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-oadd 7721  df-er 7899  df-en 8110  df-fin 8113  df-fi 8470  df-rest 16256  df-topgen 16277  df-top 20872  df-topon 20889  df-bases 20923  df-nei 21075  df-lly 21442  df-nlly 21443
This theorem is referenced by: (None)
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