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Theorem nllytop 19952
Description: A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllytop  |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )

Proof of Theorem nllytop
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlly 19948 . 2  |-  ( J  e. 𝑛Locally  A  <->  ( J  e. 
Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
21simplbi 460 1  |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1804   A.wral 2793   E.wrex 2794    i^i cin 3460   ~Pcpw 3997   {csn 4014   ` cfv 5578  (class class class)co 6281   ↾t crest 14800   Topctop 19372   neicnei 19576  𝑛Locally cnlly 19944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-nlly 19946
This theorem is referenced by:  nlly2i  19955  restnlly  19961  nllyrest  19965  nllyidm  19968  cldllycmp  19974  llycmpkgen  20031  txnlly  20116  txkgen  20131  xkococnlem  20138  xkococn  20139  cnmptkk  20162  xkofvcn  20163  cnmptk1p  20164  cnmptk2  20165  xkocnv  20293  xkohmeo  20294
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