Theorem perftop 23185

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 2740	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isperf 23180	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘∪ 𝐽) = ∪ 𝐽))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  limPtclp 23163  Perfcperf 23164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-perf 23166

This theorem is referenced by:  perfopn  23214