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Theorem llyi 21499
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 21493 . . . 4 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
21simprbi 483 . . 3 (𝐽 ∈ Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
3 pweq 4305 . . . . . . 7 (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
43ineq2d 3957 . . . . . 6 (𝑥 = 𝑈 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑈))
54rexeqdv 3284 . . . . 5 (𝑥 = 𝑈 → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
65raleqbi1dv 3285 . . . 4 (𝑥 = 𝑈 → (∀𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
76rspccva 3448 . . 3 ((∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ∧ 𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
82, 7sylan 489 . 2 ((𝐽 ∈ Locally 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
9 eleq1 2827 . . . . . . 7 (𝑦 = 𝑃 → (𝑦𝑢𝑃𝑢))
109anbi1d 743 . . . . . 6 (𝑦 = 𝑃 → ((𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1110anbi2d 742 . . . . 5 (𝑦 = 𝑃 → ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
12 anass 684 . . . . . 6 (((𝑢𝐽𝑢𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ (𝑢𝐽 ∧ (𝑢𝑈 ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
13 elin 3939 . . . . . . . 8 (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢𝐽𝑢 ∈ 𝒫 𝑈))
14 selpw 4309 . . . . . . . . 9 (𝑢 ∈ 𝒫 𝑈𝑢𝑈)
1514anbi2i 732 . . . . . . . 8 ((𝑢𝐽𝑢 ∈ 𝒫 𝑈) ↔ (𝑢𝐽𝑢𝑈))
1613, 15bitri 264 . . . . . . 7 (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢𝐽𝑢𝑈))
1716anbi1i 733 . . . . . 6 ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) ↔ ((𝑢𝐽𝑢𝑈) ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
18 3anass 1081 . . . . . . 7 ((𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢𝑈 ∧ (𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1918anbi2i 732 . . . . . 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 3186 . . 3 (𝑦 = 𝑃 → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2322rspccva 3448 . 2 ((∀𝑦𝑈𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) ∧ 𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
248, 23stoic3 1850 1 ((𝐽 ∈ Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cin 3714  wss 3715  𝒫 cpw 4302  (class class class)co 6814  t crest 16303  Topctop 20920  Locally clly 21489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-lly 21491
This theorem is referenced by:  llynlly  21502  islly2  21509  llyrest  21510  llyidm  21513  nllyidm  21514  lly1stc  21521  dislly  21522  txlly  21661  cvmlift2lem10  31622
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