Theorem nllytop 23819

Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllytop	⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)

Proof of Theorem nllytop

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnlly 23815	. 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
2	1	simplbi 502	1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ∩ cin 4092  𝒫 cpw 4560  {csn 4588  ‘cfv 6555  (class class class)co 7440   ↾_t crest 17617  Topctop 23238  neicnei 23442  𝑛-Locally cnlly 23811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-ov 7443  df-nlly 23813

This theorem is used by:  nlly2i  23822  restnlly  23828  nllyrest  23832  nllyidm  23835  cldllycmp  23841  llycmpkgen  23898  txnlly  23983  txkgen  23998  xkococnlem  24005  xkococn  24006  cnmptkk  24029  xkofvcn  24030  cnmptk1p  24031  cnmptk2  24032  xkocnv  24160  xkohmeo  24161