Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nllyi Structured version   Visualization version   GIF version

Theorem nllyi 21088
 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 21082 . . . 4 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
21simprbi 479 . . 3 (𝐽 ∈ 𝑛-Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
3 pweq 4111 . . . . . . 7 (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
43ineq2d 3776 . . . . . 6 (𝑥 = 𝑈 → (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈))
54rexeqdv 3122 . . . . 5 (𝑥 = 𝑈 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴))
65raleqbi1dv 3123 . . . 4 (𝑥 = 𝑈 → (∀𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴 ↔ ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴))
76rspccva 3281 . . 3 ((∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴)
82, 7sylan 487 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴)
9 elin 3758 . . . . . . 7 (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈))
10 sneq 4135 . . . . . . . . . 10 (𝑦 = 𝑃 → {𝑦} = {𝑃})
1110fveq2d 6107 . . . . . . . . 9 (𝑦 = 𝑃 → ((nei‘𝐽)‘{𝑦}) = ((nei‘𝐽)‘{𝑃}))
1211eleq2d 2673 . . . . . . . 8 (𝑦 = 𝑃 → (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ↔ 𝑢 ∈ ((nei‘𝐽)‘{𝑃})))
13 selpw 4115 . . . . . . . . 9 (𝑢 ∈ 𝒫 𝑈𝑢𝑈)
1413a1i 11 . . . . . . . 8 (𝑦 = 𝑃 → (𝑢 ∈ 𝒫 𝑈𝑢𝑈))
1512, 14anbi12d 743 . . . . . . 7 (𝑦 = 𝑃 → ((𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈)))
169, 15syl5bb 271 . . . . . 6 (𝑦 = 𝑃 → (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈)))
1716anbi1d 737 . . . . 5 (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴)))
18 anass 679 . . . . 5 (((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1917, 18syl6bb 275 . . . 4 (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))))
2019rexbidv2 3030 . . 3 (𝑦 = 𝑃 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2120rspccva 3281 . 2 ((∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
228, 21stoic3 1692 1 ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  neicnei 20711  𝑛-Locally cnlly 21078 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-nlly 21080 This theorem is referenced by:  nlly2i  21089  llycmpkgen  21165
 Copyright terms: Public domain W3C validator