Theorem perftop 23164

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 2737	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isperf 23159	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘∪ 𝐽) = ∪ 𝐽))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  limPtclp 23142  Perfcperf 23143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-perf 23145

This theorem is referenced by:  perfopn  23193