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Theorem cmptop 17463
Description: A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
Assertion
Ref Expression
cmptop  |-  ( J  e.  Comp  ->  J  e. 
Top )

Proof of Theorem cmptop
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  U. J  =  U. J
21iscmp 17456 . 2  |-  ( J  e.  Comp  <->  ( J  e. 
Top  /\  A. r  e.  ~P  J ( U. J  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) U. J  =  U. s ) ) )
32simplbi 448 1  |-  ( J  e.  Comp  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    i^i cin 3321   ~Pcpw 3801   U.cuni 4017   Fincfn 7112   Topctop 16963   Compccmp 17454
This theorem is referenced by:  imacmp  17465  cmpcld  17470  fiuncmp  17472  cmpfii  17477  bwth  17478  kgeni  17574  kgentopon  17575  kgencmp  17582  kgencmp2  17583  cmpkgen  17588  txcmplem1  17678  txcmp  17680  qtopcmp  17745  cmphaushmeo  17837  ptcmpfi  17850  fclscmpi  18066  alexsubALTlem1  18083  ptcmplem1  18088  ptcmpg  18093  evth  18989  evth2  18990  ordcmp  26202  locfincmp  26398  heibor1lem  26532  cmpfiiin  26765  kelac1  27152  kelac2  27154  stoweidlem28  27767  stoweidlem50  27789  stoweidlem53  27792  stoweidlem57  27796  stoweidlem59  27798  stoweidlem62  27801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803  df-uni 4018  df-cmp 17455
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