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Theorem restnlly 23490
Description: If the property 𝐴 passes to open subspaces, then a space is n-locally 𝐴 iff it is locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Hypothesis
Ref Expression
restlly.1 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
Assertion
Ref Expression
restnlly (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)
Distinct variable groups:   𝑥,𝑗,𝐴   𝜑,𝑗,𝑥

Proof of Theorem restnlly
Dummy variables 𝑘 𝑠 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nllytop 23481 . . . . . 6 (𝑘 ∈ 𝑛-Locally 𝐴𝑘 ∈ Top)
21adantl 481 . . . . 5 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Top)
3 nlly2i 23484 . . . . . . . . 9 ((𝑘 ∈ 𝑛-Locally 𝐴𝑦𝑘𝑢𝑦) → ∃𝑠 ∈ 𝒫 𝑦𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))
433adant1l 1177 . . . . . . . 8 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) → ∃𝑠 ∈ 𝒫 𝑦𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))
5 simprl 771 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑘)
6 simprr2 1223 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑠)
7 simplr 769 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑦)
87elpwid 4609 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑠𝑦)
96, 8sstrd 3994 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑦)
10 velpw 4605 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝒫 𝑦𝑥𝑦)
119, 10sylibr 234 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ 𝒫 𝑦)
125, 11elind 4200 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘 ∩ 𝒫 𝑦))
13 simprr1 1222 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑢𝑥)
14 simpll1 1213 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝜑𝑘 ∈ 𝑛-Locally 𝐴))
1514, 1simpl2im 503 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑘 ∈ Top)
16 restabs 23173 . . . . . . . . . . . . . 14 ((𝑘 ∈ Top ∧ 𝑥𝑠𝑠 ∈ 𝒫 𝑦) → ((𝑘t 𝑠) ↾t 𝑥) = (𝑘t 𝑥))
1715, 6, 7, 16syl3anc 1373 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → ((𝑘t 𝑠) ↾t 𝑥) = (𝑘t 𝑥))
18 dfss2 3969 . . . . . . . . . . . . . . . 16 (𝑥𝑠 ↔ (𝑥𝑠) = 𝑥)
196, 18sylib 218 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥𝑠) = 𝑥)
20 elrestr 17473 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ Top ∧ 𝑠 ∈ 𝒫 𝑦𝑥𝑘) → (𝑥𝑠) ∈ (𝑘t 𝑠))
2115, 7, 5, 20syl3anc 1373 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥𝑠) ∈ (𝑘t 𝑠))
2219, 21eqeltrrd 2842 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘t 𝑠))
23 eleq2 2830 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑘t 𝑠) → (𝑥𝑗𝑥 ∈ (𝑘t 𝑠)))
24 oveq1 7438 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑘t 𝑠) → (𝑗t 𝑥) = ((𝑘t 𝑠) ↾t 𝑥))
2524eleq1d 2826 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑘t 𝑠) → ((𝑗t 𝑥) ∈ 𝐴 ↔ ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴))
2623, 25imbi12d 344 . . . . . . . . . . . . . . 15 (𝑗 = (𝑘t 𝑠) → ((𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴) ↔ (𝑥 ∈ (𝑘t 𝑠) → ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴)))
2714simpld 494 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝜑)
28 restlly.1 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
2928expr 456 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝐴) → (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴))
3029ralrimiva 3146 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑗𝐴 (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴))
3127, 30syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → ∀𝑗𝐴 (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴))
32 simprr3 1224 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑘t 𝑠) ∈ 𝐴)
3326, 31, 32rspcdva 3623 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘t 𝑠) → ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴))
3422, 33mpd 15 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴)
3517, 34eqeltrrd 2842 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑘t 𝑥) ∈ 𝐴)
3612, 13, 35jca32 515 . . . . . . . . . . 11 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
3736ex 412 . . . . . . . . . 10 ((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴)) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))))
3837reximdv2 3164 . . . . . . . . 9 ((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → (∃𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
3938rexlimdva 3155 . . . . . . . 8 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) → (∃𝑠 ∈ 𝒫 𝑦𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
404, 39mpd 15 . . . . . . 7 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
41403expb 1121 . . . . . 6 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ (𝑦𝑘𝑢𝑦)) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
4241ralrimivva 3202 . . . . 5 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → ∀𝑦𝑘𝑢𝑦𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
43 islly 23476 . . . . 5 (𝑘 ∈ Locally 𝐴 ↔ (𝑘 ∈ Top ∧ ∀𝑦𝑘𝑢𝑦𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
442, 42, 43sylanbrc 583 . . . 4 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Locally 𝐴)
4544ex 412 . . 3 (𝜑 → (𝑘 ∈ 𝑛-Locally 𝐴𝑘 ∈ Locally 𝐴))
4645ssrdv 3989 . 2 (𝜑 → 𝑛-Locally 𝐴 ⊆ Locally 𝐴)
47 llyssnlly 23486 . . 3 Locally 𝐴 ⊆ 𝑛-Locally 𝐴
4847a1i 11 . 2 (𝜑 → Locally 𝐴 ⊆ 𝑛-Locally 𝐴)
4946, 48eqssd 4001 1 (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600  (class class class)co 7431  t crest 17465  Topctop 22899  Locally clly 23472  𝑛-Locally cnlly 23473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467  df-top 22900  df-nei 23106  df-lly 23474  df-nlly 23475
This theorem is referenced by:  loclly  23495  hausnlly  23501
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