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Theorem nllytop 23482
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 23478 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
21simplbi 497 1 (𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wral 3060  wrex 3069  cin 3949  𝒫 cpw 4599  {csn 4625  cfv 6560  (class class class)co 7432  t crest 17466  Topctop 22900  neicnei 23106  𝑛-Locally cnlly 23474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-nlly 23476
This theorem is referenced by:  nlly2i  23485  restnlly  23491  nllyrest  23495  nllyidm  23498  cldllycmp  23504  llycmpkgen  23561  txnlly  23646  txkgen  23661  xkococnlem  23668  xkococn  23669  cnmptkk  23692  xkofvcn  23693  cnmptk1p  23694  cnmptk2  23695  xkocnv  23823  xkohmeo  23824
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