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

Theorem nllytop 21478
Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllytop (𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)

Proof of Theorem nllytop
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlly 21474 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
21simplbi 478 1 (𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  wral 3050  wrex 3051  cin 3714  𝒫 cpw 4302  {csn 4321  cfv 6049  (class class class)co 6813  t crest 16283  Topctop 20900  neicnei 21103  𝑛-Locally cnlly 21470
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-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-nlly 21472
This theorem is referenced by:  nlly2i  21481  restnlly  21487  nllyrest  21491  nllyidm  21494  cldllycmp  21500  llycmpkgen  21557  txnlly  21642  txkgen  21657  xkococnlem  21664  xkococn  21665  cnmptkk  21688  xkofvcn  21689  cnmptk1p  21690  cnmptk2  21691  xkocnv  21819  xkohmeo  21820
  Copyright terms: Public domain W3C validator