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Theorem perftop 19530
Description: A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
Assertion
Ref Expression
perftop  |-  ( J  e. Perf  ->  J  e.  Top )

Proof of Theorem perftop
StepHypRef Expression
1 eqid 2443 . . 3  |-  U. J  =  U. J
21isperf 19525 . 2  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  U. J )  =  U. J ) )
32simplbi 460 1  |-  ( J  e. Perf  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   U.cuni 4234   ` cfv 5578   Topctop 19267   limPtclp 19508  Perfcperf 19509
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-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-perf 19511
This theorem is referenced by:  perfopn  19559
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