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Theorem perftop 20770
 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 2610 . . 3 𝐽 = 𝐽
21isperf 20765 . 2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘ 𝐽) = 𝐽))
32simplbi 475 1 (𝐽 ∈ Perf → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  limPtclp 20748  Perfcperf 20749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-perf 20751 This theorem is referenced by:  perfopn  20799
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