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Theorem restnlly 22856
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 22847 . . . . . 6 (𝑘 ∈ 𝑛-Locally 𝐴𝑘 ∈ Top)
21adantl 483 . . . . 5 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Top)
3 nlly2i 22850 . . . . . . . . 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 770 . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑦)
87elpwid 4573 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑠𝑦)
96, 8sstrd 3958 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥𝑦)
10 velpw 4569 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝒫 𝑦𝑥𝑦)
119, 10sylibr 233 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ 𝒫 𝑦)
125, 11elind 4158 . . . . . . . . . . . 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 505 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑘 ∈ Top)
16 restabs 22539 . . . . . . . . . . . . . 14 ((𝑘 ∈ Top ∧ 𝑥𝑠𝑠 ∈ 𝒫 𝑦) → ((𝑘t 𝑠) ↾t 𝑥) = (𝑘t 𝑥))
1715, 6, 7, 16syl3anc 1372 . . . . . . . . . . . . 13 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → ((𝑘t 𝑠) ↾t 𝑥) = (𝑘t 𝑥))
18 df-ss 3931 . . . . . . . . . . . . . . . 16 (𝑥𝑠 ↔ (𝑥𝑠) = 𝑥)
196, 18sylib 217 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥𝑠) = 𝑥)
20 elrestr 17318 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ Top ∧ 𝑠 ∈ 𝒫 𝑦𝑥𝑘) → (𝑥𝑠) ∈ (𝑘t 𝑠))
2115, 7, 5, 20syl3anc 1372 . . . . . . . . . . . . . . 15 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥𝑠) ∈ (𝑘t 𝑠))
2219, 21eqeltrrd 2835 . . . . . . . . . . . . . 14 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘t 𝑠))
23 eleq2 2823 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑘t 𝑠) → (𝑥𝑗𝑥 ∈ (𝑘t 𝑠)))
24 oveq1 7368 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑘t 𝑠) → (𝑗t 𝑥) = ((𝑘t 𝑠) ↾t 𝑥))
2524eleq1d 2819 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑘t 𝑠) → ((𝑗t 𝑥) ∈ 𝐴 ↔ ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴))
2623, 25imbi12d 345 . . . . . . . . . . . . . . 15 (𝑗 = (𝑘t 𝑠) → ((𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴) ↔ (𝑥 ∈ (𝑘t 𝑠) → ((𝑘t 𝑠) ↾t 𝑥) ∈ 𝐴)))
2714simpld 496 . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝐴) → (𝑥𝑗 → (𝑗t 𝑥) ∈ 𝐴))
3029ralrimiva 3140 . . . . . . . . . . . . . . . 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 3584 . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . 12 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑘t 𝑥) ∈ 𝐴)
3612, 13, 35jca32 517 . . . . . . . . . . 11 (((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
3736ex 414 . . . . . . . . . 10 ((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑥𝑘 ∧ (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴)) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))))
3837reximdv2 3158 . . . . . . . . 9 ((((𝜑𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦𝑘𝑢𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → (∃𝑥𝑘 (𝑢𝑥𝑥𝑠 ∧ (𝑘t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
3938rexlimdva 3149 . . . . . . . 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 3194 . . . . 5 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → ∀𝑦𝑘𝑢𝑦𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴))
43 islly 22842 . . . . 5 (𝑘 ∈ Locally 𝐴 ↔ (𝑘 ∈ Top ∧ ∀𝑦𝑘𝑢𝑦𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢𝑥 ∧ (𝑘t 𝑥) ∈ 𝐴)))
442, 42, 43sylanbrc 584 . . . 4 ((𝜑𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Locally 𝐴)
4544ex 414 . . 3 (𝜑 → (𝑘 ∈ 𝑛-Locally 𝐴𝑘 ∈ Locally 𝐴))
4645ssrdv 3954 . 2 (𝜑 → 𝑛-Locally 𝐴 ⊆ Locally 𝐴)
47 llyssnlly 22852 . . 3 Locally 𝐴 ⊆ 𝑛-Locally 𝐴
4847a1i 11 . 2 (𝜑 → Locally 𝐴 ⊆ 𝑛-Locally 𝐴)
4946, 48eqssd 3965 1 (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cin 3913  wss 3914  𝒫 cpw 4564  (class class class)co 7361  t crest 17310  Topctop 22265  Locally clly 22838  𝑛-Locally cnlly 22839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-rest 17312  df-top 22266  df-nei 22472  df-lly 22840  df-nlly 22841
This theorem is referenced by:  loclly  22861  hausnlly  22867
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