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Theorem nllytop 19089
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 19085 . 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 1756   A.wral 2727   E.wrex 2728    i^i cin 3339   ~Pcpw 3872   {csn 3889   ` cfv 5430  (class class class)co 6103   ↾t crest 14371   Topctop 18510   neicnei 18713  𝑛Locally cnlly 19081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-iota 5393  df-fv 5438  df-ov 6106  df-nlly 19083
This theorem is referenced by:  nlly2i  19092  restnlly  19098  nllyrest  19102  nllyidm  19105  cldllycmp  19111  llycmpkgen  19137  txnlly  19222  txkgen  19237  xkococnlem  19244  xkococn  19245  cnmptkk  19268  xkofvcn  19269  cnmptk1p  19270  cnmptk2  19271  xkocnv  19399  xkohmeo  19400
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