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

Theorem nllyi 22165
 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 22159 . . . 4 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
21simprbi 501 . . 3 (𝐽 ∈ 𝑛-Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
3 pweq 4508 . . . . . . 7 (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
43ineq2d 4118 . . . . . 6 (𝑥 = 𝑈 → (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈))
54rexeqdv 3331 . . . . 5 (𝑥 = 𝑈 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴))
65raleqbi1dv 3322 . . . 4 (𝑥 = 𝑈 → (∀𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴 ↔ ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴))
76rspccva 3541 . . 3 ((∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴)
82, 7sylan 584 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽) → ∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴)
9 elin 3875 . . . . . . 7 (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈))
10 sneq 4530 . . . . . . . . . 10 (𝑦 = 𝑃 → {𝑦} = {𝑃})
1110fveq2d 6660 . . . . . . . . 9 (𝑦 = 𝑃 → ((nei‘𝐽)‘{𝑦}) = ((nei‘𝐽)‘{𝑃}))
1211eleq2d 2838 . . . . . . . 8 (𝑦 = 𝑃 → (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ↔ 𝑢 ∈ ((nei‘𝐽)‘{𝑃})))
13 velpw 4497 . . . . . . . . 9 (𝑢 ∈ 𝒫 𝑈𝑢𝑈)
1413a1i 11 . . . . . . . 8 (𝑦 = 𝑃 → (𝑢 ∈ 𝒫 𝑈𝑢𝑈))
1512, 14anbi12d 634 . . . . . . 7 (𝑦 = 𝑃 → ((𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈)))
169, 15syl5bb 286 . . . . . 6 (𝑦 = 𝑃 → (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈)))
1716anbi1d 633 . . . . 5 (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ ((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴)))
18 anass 473 . . . . 5 (((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴)))
1917, 18bitrdi 290 . . . 4 (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))))
2019rexbidv2 3220 . . 3 (𝑦 = 𝑃 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴)))
2120rspccva 3541 . 2 ((∀𝑦𝑈𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽t 𝑢) ∈ 𝐴𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
228, 21stoic3 1779 1 ((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4492  {csn 4520  ‘cfv 6333  (class class class)co 7148   ↾t crest 16742  Topctop 21583  neicnei 21787  𝑛-Locally cnlly 22155 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-un 3864  df-in 3866  df-ss 3876  df-pw 4494  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-br 5031  df-iota 6292  df-fv 6341  df-ov 7151  df-nlly 22157 This theorem is referenced by:  nlly2i  22166  llycmpkgen  22242
 Copyright terms: Public domain W3C validator