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Theorem nllytop 19737
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 19733 . 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 1767   A.wral 2814   E.wrex 2815    i^i cin 3475   ~Pcpw 4010   {csn 4027   ` cfv 5586  (class class class)co 6282   ↾t crest 14669   Topctop 19158   neicnei 19361  𝑛Locally cnlly 19729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-nlly 19731
This theorem is referenced by:  nlly2i  19740  restnlly  19746  nllyrest  19750  nllyidm  19753  cldllycmp  19759  llycmpkgen  19785  txnlly  19870  txkgen  19885  xkococnlem  19892  xkococn  19893  cnmptkk  19916  xkofvcn  19917  cnmptk1p  19918  cnmptk2  19919  xkocnv  20047  xkohmeo  20048
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