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Theorem restnlly 22018
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 22009 . . . . . 6 (𝑘 ∈ 𝑛-Locally 𝐴𝑘 ∈ Top)
21adantl 482 . . . . 5 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Top)
3 nlly2i 22012 . . . . . . . . 9 ((𝑘 ∈ 𝑛-Locally 𝐴𝑦𝑘𝑢𝑦) → ∃𝑠 ∈ 𝒫 𝑦𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))
433adant1l 1168 . . . . . . . 8 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) → ∃𝑠 ∈ 𝒫 𝑦𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))
5 simprl 767 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑘)
6 simprr2 1214 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑠)
7 simplr 765 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑦)
87elpwid 4549 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑠𝑦)
96, 8sstrd 3974 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑦)
10 velpw 4543 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝒫 𝑦𝑥𝑦)
119, 10sylibr 235 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ 𝒫 𝑦)
125, 11elind 4168 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘 ∩ 𝒫 𝑦))
13 simprr1 1213 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑢𝑥)
14 simpll1 1204 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝜑𝑘 ∈ 𝑛-Locally 𝐴))
1514, 1simpl2im 504 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑘 ∈ Top)
16 restabs 21701 . . . . . . . . . . . . . 14 ((𝑘 ∈ Top ∧ 𝑥𝑠𝑠 ∈ 𝒫 𝑦) → ((𝑘t 𝑠) ↾t 𝑥) = (𝑘t 𝑥))
1715, 6, 7, 16syl3anc 1363 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → ((𝑘t 𝑠) ↾t 𝑥) = (𝑘t 𝑥))
18 df-ss 3949 . . . . . . . . . . . . . . . 16 (𝑥𝑠 ↔ (𝑥𝑠) = 𝑥)
196, 18sylib 219 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥𝑠) = 𝑥)
20 elrestr 16690 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ Top ∧ 𝑠 ∈ 𝒫 𝑦𝑥𝑘) → (𝑥𝑠) ∈ (𝑘t 𝑠))
2115, 7, 5, 20syl3anc 1363 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥𝑠) ∈ (𝑘t 𝑠))
2219, 21eqeltrrd 2911 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘t 𝑠))
23 eleq2 2898 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑘t 𝑠) → (𝑥𝑗𝑥 ∈ (𝑘t 𝑠)))
24 oveq1 7152 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑘t 𝑠) → (𝑗t 𝑥) = ((𝑘t 𝑠) ↾t 𝑥))
2524eleq1d 2894 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑘t 𝑠) → ((𝑗t 𝑥) ∈ 𝐴 ↔ ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴))
2623, 25imbi12d 346 . . . . . . . . . . . . . . 15 (𝑗 = (𝑘t 𝑠) → ((𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴) ↔ (𝑥 ∈ (𝑘t 𝑠) → ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴)))
2714simpld 495 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝐴) → (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴))
3029ralrimiva 3179 . . . . . . . . . . . . . . . 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 1215 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑘t 𝑠) ∈ 𝐴)
3326, 31, 32rspcdva 3622 . . . . . . . . . . . . . 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 2911 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑘t 𝑥) ∈ 𝐴)
3612, 13, 35jca32 516 . . . . . . . . . . 11 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
3736ex 413 . . . . . . . . . 10 ((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴)) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))))
3837reximdv2 3268 . . . . . . . . 9 ((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → (∃𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
3938rexlimdva 3281 . . . . . . . 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 1112 . . . . . 6 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ (𝑦𝑘𝑢𝑦)) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
4241ralrimivva 3188 . . . . 5 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → ∀𝑦𝑘𝑢𝑦𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
43 islly 22004 . . . . 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 413 . . 3 (𝜑 → (𝑘 ∈ 𝑛-Locally 𝐴𝑘 ∈ Locally 𝐴))
4645ssrdv 3970 . 2 (𝜑 → 𝑛-Locally 𝐴 ⊆ Locally 𝐴)
47 llyssnlly 22014 . . 3 Locally 𝐴 ⊆ 𝑛-Locally 𝐴
4847a1i 11 . 2 (𝜑 → Locally 𝐴 ⊆ 𝑛-Locally 𝐴)
4946, 48eqssd 3981 1 (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cin 3932  wss 3933  𝒫 cpw 4535  (class class class)co 7145  t crest 16682  Topctop 21429  Locally clly 22000  𝑛-Locally cnlly 22001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rest 16684  df-top 21430  df-nei 21634  df-lly 22002  df-nlly 22003
This theorem is referenced by:  loclly  22023  hausnlly  22029
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