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Theorem nllyidm 22013
Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 22011 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 21996 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴𝑗 ∈ Top)
2 llyi 21998 . . . . . . 7 ((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑢𝑗 (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))
3 simprr3 1217 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴)
4 simprl 767 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑗)
5 ssidd 3993 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑢)
6 simpl1 1185 . . . . . . . . . . . 12 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Locally 𝑛-Locally 𝐴)
76, 1syl 17 . . . . . . . . . . 11 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Top)
8 restopn2 21701 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
97, 4, 8syl2anc 584 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
104, 5, 9mpbir2and 709 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ∈ (𝑗t 𝑢))
11 simprr2 1216 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑦𝑢)
12 nlly2i 22000 . . . . . . . . 9 (((𝑗t 𝑢) ∈ 𝑛-Locally 𝐴𝑢 ∈ (𝑗t 𝑢) ∧ 𝑦𝑢) → ∃𝑣 ∈ 𝒫 𝑢𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
133, 10, 11, 12syl3anc 1365 . . . . . . . 8 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ 𝒫 𝑢𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
14 restopn2 21701 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
157, 4, 14syl2anc 584 . . . . . . . . . . . . 13 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
1615adantr 481 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
177adantr 481 . . . . . . . . . . . . . . . . 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)
18 simpr2l 1226 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧𝑗)
19 simpr31 1257 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦𝑧)
20 opnneip 21643 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑧𝑗𝑦𝑧) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
2117, 18, 19, 20syl3anc 1365 . . . . . . . . . . . . . . . . 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}))
22 simpr32 1258 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧𝑣)
23 simpr1 1188 . . . . . . . . . . . . . . . . . . 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)
2423elpwid 4555 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑢)
254adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑗)
26 elssuni 4865 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑗𝑢 𝑗)
2725, 26syl 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)
2824, 27sstrd 3980 . . . . . . . . . . . . . . . . 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)
29 eqid 2825 . . . . . . . . . . . . . . . . . 18 𝑗 = 𝑗
3029ssnei2 21640 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ Top ∧ 𝑧 ∈ ((nei‘𝑗)‘{𝑦})) ∧ (𝑧𝑣𝑣 𝑗)) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
3117, 21, 22, 28, 30syl22anc 836 . . . . . . . . . . . . . . . 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}))
32 simprr1 1215 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑥)
3332adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑥)
3424, 33sstrd 3980 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
35 velpw 4549 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
3634, 35sylibr 235 . . . . . . . . . . . . . . . 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)
3731, 36elind 4174 . . . . . . . . . . . . . . 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))
38 restabs 21689 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ Top ∧ 𝑣𝑢𝑢𝑗) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
3917, 24, 25, 38syl3anc 1365 . . . . . . . . . . . . . . . 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))
40 simpr33 1259 . . . . . . . . . . . . . . . 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)
4139, 40eqeltrrd 2918 . . . . . . . . . . . . . . 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)
4237, 41jca 512 . . . . . . . . . . . . . 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))
43423exp2 1348 . . . . . . . . . . . . 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)))))
4443imp 407 . . . . . . . . . . . 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))))
4516, 44sylbid 241 . . . . . . . . . . 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))))
4645rexlimdv 3287 . . . . . . . . . 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)))
4746expimpd 454 . . . . . . . . 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)))
4847reximdv2 3275 . . . . . . . 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))
4913, 48mpd 15 . . . . . . 7 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
502, 49rexlimddv 3295 . . . . . 6 ((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
51503expb 1114 . . . . 5 ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
5251ralrimivva 3195 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴 → ∀𝑥𝑗𝑦𝑥𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
53 isnlly 21993 . . . 4 (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴))
541, 52, 53sylanbrc 583 . . 3 (𝑗 ∈ Locally 𝑛-Locally 𝐴𝑗 ∈ 𝑛-Locally 𝐴)
5554ssriv 3974 . 2 Locally 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐴
56 nllyrest 22010 . . . . 5 ((𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
5756adantl 482 . . . 4 ((⊤ ∧ (𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
58 nllytop 21997 . . . . . 6 (𝑗 ∈ 𝑛-Locally 𝐴𝑗 ∈ Top)
5958ssriv 3974 . . . . 5 𝑛-Locally 𝐴 ⊆ Top
6059a1i 11 . . . 4 (⊤ → 𝑛-Locally 𝐴 ⊆ Top)
6157, 60restlly 22007 . . 3 (⊤ → 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴)
6261mptru 1537 . 2 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴
6355, 62eqssi 3986 1 Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wtru 1531  wcel 2107  wral 3142  wrex 3143  cin 3938  wss 3939  𝒫 cpw 4541  {csn 4563   cuni 4836  cfv 6351  (class class class)co 7151  t crest 16686  Topctop 21417  neicnei 21621  Locally clly 21988  𝑛-Locally cnlly 21989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100  df-er 8282  df-en 8502  df-fin 8505  df-fi 8867  df-rest 16688  df-topgen 16709  df-top 21418  df-topon 21435  df-bases 21470  df-nei 21622  df-lly 21990  df-nlly 21991
This theorem is referenced by: (None)
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