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Theorem llyi 21258
Description: The property of a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyi ((𝐽 ∈ Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑃   𝑢,𝑈   𝑢,𝐽

Proof of Theorem llyi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islly 21252 . . . 4 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
21simprbi 480 . . 3 (𝐽 ∈ Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
3 pweq 4152 . . . . . . 7 (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
43ineq2d 3806 . . . . . 6 (𝑥 = 𝑈 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑈))
54rexeqdv 3140 . . . . 5 (𝑥 = 𝑈 → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
65raleqbi1dv 3141 . . . 4 (𝑥 = 𝑈 → (∀𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
76rspccva 3303 . . 3 ((∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ∧ 𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
82, 7sylan 488 . 2 ((𝐽 ∈ Locally 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
9 eleq1 2687 . . . . . . 7 (𝑦 = 𝑃 → (𝑦𝑢𝑃𝑢))
109anbi1d 740 . . . . . 6 (𝑦 = 𝑃 → ((𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1110anbi2d 739 . . . . 5 (𝑦 = 𝑃 → ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
12 anass 680 . . . . . 6 (((𝑢𝐽𝑢𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈 ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
13 elin 3788 . . . . . . . 8 (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢𝐽𝑢 ∈ 𝒫 𝑈))
14 selpw 4156 . . . . . . . . 9 (𝑢 ∈ 𝒫 𝑈𝑢𝑈)
1514anbi2i 729 . . . . . . . 8 ((𝑢𝐽𝑢 ∈ 𝒫 𝑈) ↔ (𝑢𝐽𝑢𝑈))
1613, 15bitri 264 . . . . . . 7 (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢𝐽𝑢𝑈))
1716anbi1i 730 . . . . . 6 ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ ((𝑢𝐽𝑢𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
18 3anass 1040 . . . . . . 7 ((𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢𝑈 ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1918anbi2i 729 . . . . . 6 ((𝑢𝐽 ∧ (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈 ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
2012, 17, 193bitr4i 292 . . . . 5 ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2111, 20syl6bb 276 . . . 4 (𝑦 = 𝑃 → ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
2221rexbidv2 3044 . . 3 (𝑦 = 𝑃 → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2322rspccva 3303 . 2 ((∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ∧ 𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
248, 23stoic3 1699 1 ((𝐽 ∈ Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wrex 2910  cin 3566  wss 3567  𝒫 cpw 4149  (class class class)co 6635  t crest 16062  Topctop 20679  Locally clly 21248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-lly 21250
This theorem is referenced by:  llynlly  21261  islly2  21268  llyrest  21269  llyidm  21272  nllyidm  21273  lly1stc  21280  dislly  21281  txlly  21420  cvmlift2lem10  31268
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