Theorem nllytop 23792

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 23788	. 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 2144  ∀wral 3076  ∃wrex 3086   ∩ cin 4089  𝒫 cpw 4556  {csn 4583  ‘cfv 6538  (class class class)co 7420   ↾_t crest 17591  Topctop 23211  neicnei 23415  𝑛-Locally cnlly 23784

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-nlly 23786

This theorem is used by:  nlly2i  23795  restnlly  23801  nllyrest  23805  nllyidm  23808  cldllycmp  23814  llycmpkgen  23871  txnlly  23956  txkgen  23971  xkococnlem  23978  xkococn  23979  cnmptkk  24002  xkofvcn  24003  cnmptk1p  24004  cnmptk2  24005  xkocnv  24133  xkohmeo  24134