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Theorem cmptop 21919
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 2826 . . 3 𝐽 = 𝐽
21iscmp 21912 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑠)))
32simplbi 498 1 (𝐽 ∈ Comp → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  wral 3143  wrex 3144  cin 3939  𝒫 cpw 4542   cuni 4837  Fincfn 8498  Topctop 21417  Compccmp 21910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-in 3947  df-ss 3956  df-pw 4544  df-uni 4838  df-cmp 21911
This theorem is referenced by:  imacmp  21921  cmpcld  21926  fiuncmp  21928  cmpfii  21933  bwth  21934  locfincmp  22050  kgeni  22061  kgentopon  22062  kgencmp  22069  kgencmp2  22070  cmpkgen  22075  txcmplem1  22165  txcmp  22167  qtopcmp  22232  cmphaushmeo  22324  ptcmpfi  22337  fclscmpi  22553  alexsubALTlem1  22571  ptcmplem1  22576  ptcmpg  22581  evth  23478  evth2  23479  cmppcmp  31008  ordcmp  33679  poimirlem30  34789  heibor1lem  34955  cmpfiiin  39159  kelac1  39528  kelac2  39530  stoweidlem28  42179  stoweidlem50  42201  stoweidlem53  42204  stoweidlem57  42208  stoweidlem62  42213
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