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Theorem cmptop 22769
Description: A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
Assertion
Ref Expression
cmptop (𝐽 ∈ Comp → 𝐽 ∈ Top)

Proof of Theorem cmptop
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 𝐽 = 𝐽
21iscmp 22762 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑠)))
32simplbi 499 1 (𝐽 ∈ Comp → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cin 3913  𝒫 cpw 4564   cuni 4869  Fincfn 8889  Topctop 22265  Compccmp 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566  df-uni 4870  df-cmp 22761
This theorem is referenced by:  imacmp  22771  cmpcld  22776  fiuncmp  22778  cmpfii  22783  bwth  22784  locfincmp  22900  kgeni  22911  kgentopon  22912  kgencmp  22919  kgencmp2  22920  cmpkgen  22925  txcmplem1  23015  txcmp  23017  qtopcmp  23082  cmphaushmeo  23174  ptcmpfi  23187  fclscmpi  23403  alexsubALTlem1  23421  ptcmplem1  23426  ptcmpg  23431  evth  24345  evth2  24346  cmppcmp  32503  ordcmp  34972  poimirlem30  36158  heibor1lem  36318  cmpfiiin  41067  kelac1  41437  kelac2  41439  stoweidlem28  44359  stoweidlem50  44381  stoweidlem53  44384  stoweidlem57  44388  stoweidlem62  44393
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