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Theorem llyi 22371
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 22365 . . . 4 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
21simprbi 500 . . 3 (𝐽 ∈ Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
3 pweq 4529 . . . . . . 7 (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
43ineq2d 4127 . . . . . 6 (𝑥 = 𝑈 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑈))
54rexeqdv 3326 . . . . 5 (𝑥 = 𝑈 → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
65raleqbi1dv 3317 . . . 4 (𝑥 = 𝑈 → (∀𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
76rspccva 3536 . . 3 ((∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ∧ 𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
82, 7sylan 583 . 2 ((𝐽 ∈ Locally 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
9 eleq1 2825 . . . . . . 7 (𝑦 = 𝑃 → (𝑦𝑢𝑃𝑢))
109anbi1d 633 . . . . . 6 (𝑦 = 𝑃 → ((𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1110anbi2d 632 . . . . 5 (𝑦 = 𝑃 → ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
12 anass 472 . . . . . 6 (((𝑢𝐽𝑢𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈 ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
13 elin 3882 . . . . . . . 8 (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢𝐽𝑢 ∈ 𝒫 𝑈))
14 velpw 4518 . . . . . . . . 9 (𝑢 ∈ 𝒫 𝑈𝑢𝑈)
1514anbi2i 626 . . . . . . . 8 ((𝑢𝐽𝑢 ∈ 𝒫 𝑈) ↔ (𝑢𝐽𝑢𝑈))
1613, 15bitri 278 . . . . . . 7 (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢𝐽𝑢𝑈))
1716anbi1i 627 . . . . . 6 ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ ((𝑢𝐽𝑢𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
18 3anass 1097 . . . . . . 7 ((𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢𝑈 ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1918anbi2i 626 . . . . . 6 ((𝑢𝐽 ∧ (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈 ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
2012, 17, 193bitr4i 306 . . . . 5 ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2111, 20bitrdi 290 . . . 4 (𝑦 = 𝑃 → ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
2221rexbidv2 3214 . . 3 (𝑦 = 𝑃 → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2322rspccva 3536 . 2 ((∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ∧ 𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
248, 23stoic3 1784 1 ((𝐽 ∈ Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  cin 3865  wss 3866  𝒫 cpw 4513  (class class class)co 7213  t crest 16925  Topctop 21790  Locally clly 22361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-lly 22363
This theorem is referenced by:  llynlly  22374  islly2  22381  llyrest  22382  llyidm  22385  nllyidm  22386  lly1stc  22393  dislly  22394  txlly  22533  cvmlift2lem10  32987
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