Theorem perftop 23104

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

Step	Hyp	Ref	Expression
1		eqid 2725	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isperf 23099	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘∪ 𝐽) = ∪ 𝐽))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∪ cuni 4909  ‘cfv 6549  Topctop 22839  limPtclp 23082  Perfcperf 23083

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-perf 23085

This theorem is referenced by:  perfopn  23133