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Theorem nllytop 22074
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 22070 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
21simplbi 501 1 (𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  wral 3133  wrex 3134  cin 3918  𝒫 cpw 4521  {csn 4549  cfv 6343  (class class class)co 7145  t crest 16690  Topctop 21494  neicnei 21698  𝑛-Locally cnlly 22066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7148  df-nlly 22068
This theorem is referenced by:  nlly2i  22077  restnlly  22083  nllyrest  22087  nllyidm  22090  cldllycmp  22096  llycmpkgen  22153  txnlly  22238  txkgen  22253  xkococnlem  22260  xkococn  22261  cnmptkk  22284  xkofvcn  22285  cnmptk1p  22286  cnmptk2  22287  xkocnv  22415  xkohmeo  22416
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