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Theorem nllyi 23366
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 23360 . . . 4 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
21simprbi 496 . . 3 (𝐽 ∈ 𝑛-Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
3 pweq 4612 . . . . . . 7 (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
43ineq2d 4208 . . . . . 6 (𝑥 = 𝑈 → (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈))
54rexeqdv 3321 . . . . 5 (𝑥 = 𝑈 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴))
65raleqbi1dv 3328 . . . 4 (𝑥 = 𝑈 → (∀𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴 ↔ ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴))
76rspccva 3606 . . 3 ((∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴)
82, 7sylan 579 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴)
9 elin 3960 . . . . . . 7 (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈))
10 sneq 4634 . . . . . . . . . 10 (𝑦 = 𝑃 → {𝑦} = {𝑃})
1110fveq2d 6895 . . . . . . . . 9 (𝑦 = 𝑃 → ((nei‘𝐽)‘{𝑦}) = ((nei‘𝐽)‘{𝑃}))
1211eleq2d 2814 . . . . . . . 8 (𝑦 = 𝑃 → (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ↔ 𝑢 ∈ ((nei‘𝐽)‘{𝑃})))
13 velpw 4603 . . . . . . . . 9 (𝑢 ∈ 𝒫 𝑈𝑢𝑈)
1413a1i 11 . . . . . . . 8 (𝑦 = 𝑃 → (𝑢 ∈ 𝒫 𝑈𝑢𝑈))
1512, 14anbi12d 630 . . . . . . 7 (𝑦 = 𝑃 → ((𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈)))
169, 15bitrid 283 . . . . . 6 (𝑦 = 𝑃 → (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈)))
1716anbi1d 629 . . . . 5 (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴)))
18 anass 468 . . . . 5 (((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1917, 18bitrdi 287 . . . 4 (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))))
2019rexbidv2 3169 . . 3 (𝑦 = 𝑃 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2120rspccva 3606 . 2 ((∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
228, 21stoic3 1771 1 ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056  wrex 3065  cin 3943  wss 3944  𝒫 cpw 4598  {csn 4624  cfv 6542  (class class class)co 7414  t crest 17393  Topctop 22782  neicnei 22988  𝑛-Locally cnlly 23356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-nlly 23358
This theorem is referenced by:  nlly2i  23367  llycmpkgen  23443
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