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Theorem nllyi 21478
Description: The property of an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyi ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑃   𝑢,𝑈   𝑢,𝐽

Proof of Theorem nllyi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlly 21472 . . . 4 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
21simprbi 483 . . 3 (𝐽 ∈ 𝑛-Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
3 pweq 4303 . . . . . . 7 (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
43ineq2d 3955 . . . . . 6 (𝑥 = 𝑈 → (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈))
54rexeqdv 3282 . . . . 5 (𝑥 = 𝑈 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴))
65raleqbi1dv 3283 . . . 4 (𝑥 = 𝑈 → (∀𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴 ↔ ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴))
76rspccva 3446 . . 3 ((∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴)
82, 7sylan 489 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴)
9 elin 3937 . . . . . . 7 (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈))
10 sneq 4329 . . . . . . . . . 10 (𝑦 = 𝑃 → {𝑦} = {𝑃})
1110fveq2d 6354 . . . . . . . . 9 (𝑦 = 𝑃 → ((nei‘𝐽)‘{𝑦}) = ((nei‘𝐽)‘{𝑃}))
1211eleq2d 2823 . . . . . . . 8 (𝑦 = 𝑃 → (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ↔ 𝑢 ∈ ((nei‘𝐽)‘{𝑃})))
13 selpw 4307 . . . . . . . . 9 (𝑢 ∈ 𝒫 𝑈𝑢𝑈)
1413a1i 11 . . . . . . . 8 (𝑦 = 𝑃 → (𝑢 ∈ 𝒫 𝑈𝑢𝑈))
1512, 14anbi12d 749 . . . . . . 7 (𝑦 = 𝑃 → ((𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈)))
169, 15syl5bb 272 . . . . . 6 (𝑦 = 𝑃 → (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈)))
1716anbi1d 743 . . . . 5 (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴)))
18 anass 684 . . . . 5 (((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1917, 18syl6bb 276 . . . 4 (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))))
2019rexbidv2 3184 . . 3 (𝑦 = 𝑃 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2120rspccva 3446 . 2 ((∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
228, 21stoic3 1848 1 ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  wrex 3049  cin 3712  wss 3713  𝒫 cpw 4300  {csn 4319  cfv 6047  (class class class)co 6811  t crest 16281  Topctop 20898  neicnei 21101  𝑛-Locally cnlly 21468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-iota 6010  df-fv 6055  df-ov 6814  df-nlly 21470
This theorem is referenced by:  nlly2i  21479  llycmpkgen  21555
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