Theorem nllytop 22083

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 22079	. 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
2	1	simplbi 500	1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ∩ cin 3937  𝒫 cpw 4541  {csn 4569  ‘cfv 6357  (class class class)co 7158   ↾_t crest 16696  Topctop 21503  neicnei 21707  𝑛-Locally cnlly 22075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-nlly 22077

This theorem is referenced by:  nlly2i  22086  restnlly  22092  nllyrest  22096  nllyidm  22099  cldllycmp  22105  llycmpkgen  22162  txnlly  22247  txkgen  22262  xkococnlem  22269  xkococn  22270  cnmptkk  22293  xkofvcn  22294  cnmptk1p  22295  cnmptk2  22296  xkocnv  22424  xkohmeo  22425