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Theorem restnlly 21204
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 21195 . . . . . 6 (𝑘 ∈ 𝑛-Locally 𝐴𝑘 ∈ Top)
21adantl 482 . . . . 5 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Top)
3 nlly2i 21198 . . . . . . . . 9 ((𝑘 ∈ 𝑛-Locally 𝐴𝑦𝑘𝑢𝑦) → ∃𝑠 ∈ 𝒫 𝑦𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))
433adant1l 1315 . . . . . . . 8 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) → ∃𝑠 ∈ 𝒫 𝑦𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))
5 simprl 793 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑘)
6 simprr2 1108 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑠)
7 simplr 791 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑦)
87elpwid 4146 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑠𝑦)
96, 8sstrd 3597 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑦)
10 selpw 4142 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝒫 𝑦𝑥𝑦)
119, 10sylibr 224 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ 𝒫 𝑦)
125, 11elind 3781 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘 ∩ 𝒫 𝑦))
13 simprr1 1107 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑢𝑥)
14 simpll1 1098 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝜑𝑘 ∈ 𝑛-Locally 𝐴))
1514simprd 479 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑘 ∈ 𝑛-Locally 𝐴)
1615, 1syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑘 ∈ Top)
17 restabs 20888 . . . . . . . . . . . . . 14 ((𝑘 ∈ Top ∧ 𝑥𝑠𝑠 ∈ 𝒫 𝑦) → ((𝑘t 𝑠) ↾t 𝑥) = (𝑘t 𝑥))
1816, 6, 7, 17syl3anc 1323 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → ((𝑘t 𝑠) ↾t 𝑥) = (𝑘t 𝑥))
19 simprr3 1109 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑘t 𝑠) ∈ 𝐴)
2014simpld 475 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝜑)
21 restlly.1 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
2221expr 642 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝐴) → (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴))
2322ralrimiva 2961 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑗𝐴 (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴))
2420, 23syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → ∀𝑗𝐴 (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴))
25 df-ss 3573 . . . . . . . . . . . . . . . 16 (𝑥𝑠 ↔ (𝑥𝑠) = 𝑥)
266, 25sylib 208 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥𝑠) = 𝑥)
27 elrestr 16017 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ Top ∧ 𝑠 ∈ 𝒫 𝑦𝑥𝑘) → (𝑥𝑠) ∈ (𝑘t 𝑠))
2816, 7, 5, 27syl3anc 1323 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥𝑠) ∈ (𝑘t 𝑠))
2926, 28eqeltrrd 2699 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘t 𝑠))
30 eleq2 2687 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑘t 𝑠) → (𝑥𝑗𝑥 ∈ (𝑘t 𝑠)))
31 oveq1 6617 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑘t 𝑠) → (𝑗t 𝑥) = ((𝑘t 𝑠) ↾t 𝑥))
3231eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑘t 𝑠) → ((𝑗t 𝑥) ∈ 𝐴 ↔ ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴))
3330, 32imbi12d 334 . . . . . . . . . . . . . . 15 (𝑗 = (𝑘t 𝑠) → ((𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴) ↔ (𝑥 ∈ (𝑘t 𝑠) → ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴)))
3433rspcv 3294 . . . . . . . . . . . . . 14 ((𝑘t 𝑠) ∈ 𝐴 → (∀𝑗𝐴 (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴) → (𝑥 ∈ (𝑘t 𝑠) → ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴)))
3519, 24, 29, 34syl3c 66 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴)
3618, 35eqeltrrd 2699 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑘t 𝑥) ∈ 𝐴)
3712, 13, 36jca32 557 . . . . . . . . . . 11 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
3837ex 450 . . . . . . . . . 10 ((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴)) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))))
3938reximdv2 3009 . . . . . . . . 9 ((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → (∃𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
4039rexlimdva 3025 . . . . . . . 8 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) → (∃𝑠 ∈ 𝒫 𝑦𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
414, 40mpd 15 . . . . . . 7 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
42413expb 1263 . . . . . 6 (((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ (𝑦𝑘𝑢𝑦)) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
4342ralrimivva 2966 . . . . 5 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → ∀𝑦𝑘𝑢𝑦𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
44 islly 21190 . . . . 5 (𝑘 ∈ Locally 𝐴 ↔ (𝑘 ∈ Top ∧ ∀𝑦𝑘𝑢𝑦𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
452, 43, 44sylanbrc 697 . . . 4 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Locally 𝐴)
4645ex 450 . . 3 (𝜑 → (𝑘 ∈ 𝑛-Locally 𝐴𝑘 ∈ Locally 𝐴))
4746ssrdv 3593 . 2 (𝜑 → 𝑛-Locally 𝐴 ⊆ Locally 𝐴)
48 llyssnlly 21200 . . 3 Locally 𝐴 ⊆ 𝑛-Locally 𝐴
4948a1i 11 . 2 (𝜑 → Locally 𝐴 ⊆ 𝑛-Locally 𝐴)
5047, 49eqssd 3604 1 (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3558  wss 3559  𝒫 cpw 4135  (class class class)co 6610  t crest 16009  Topctop 20626  Locally clly 21186  𝑛-Locally cnlly 21187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-rest 16011  df-top 20627  df-nei 20821  df-lly 21188  df-nlly 21189
This theorem is referenced by:  loclly  21209  hausnlly  21215
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