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Theorem cmptop 17138
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 2296 . . 3  |-  U. J  =  U. J
21iscmp 17131 . 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 446 1  |-  ( J  e.  Comp  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164   ~Pcpw 3638   U.cuni 3843   Fincfn 6879   Topctop 16647   Compccmp 17129
This theorem is referenced by:  imacmp  17140  cmpcld  17145  fiuncmp  17147  cmpfii  17152  kgeni  17248  kgentopon  17249  kgencmp  17256  kgencmp2  17257  cmpkgen  17262  txcmplem1  17351  txcmp  17353  qtopcmp  17415  cmphaushmeo  17507  ptcmpfi  17520  fclscmpi  17740  alexsubALTlem1  17757  ptcmplem1  17762  ptcmpg  17767  evth  18473  evth2  18474  ordcmp  24958  bwt2  25695  locfincmp  26407  heibor1lem  26636  cmpfiiin  26875  kelac1  27264  kelac2  27266  stoweidlem28  27880  stoweidlem50  27902  stoweidlem53  27905  stoweidlem57  27909  stoweidlem59  27911  stoweidlem62  27914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-cmp 17130
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