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Theorem perftop 21764
Description: A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
Assertion
Ref Expression
perftop (𝐽 ∈ Perf → 𝐽 ∈ Top)

Proof of Theorem perftop
StepHypRef Expression
1 eqid 2824 . . 3 𝐽 = 𝐽
21isperf 21759 . 2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘ 𝐽) = 𝐽))
32simplbi 501 1 (𝐽 ∈ Perf → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115   cuni 4824  cfv 6343  Topctop 21501  limPtclp 21742  Perfcperf 21743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-perf 21745
This theorem is referenced by:  perfopn  21793
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