Theorem perftop 23071

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 2731	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isperf 23066	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘∪ 𝐽) = ∪ 𝐽))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∪ cuni 4856  ‘cfv 6481  Topctop 22808  limPtclp 23049  Perfcperf 23050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-perf 23052

This theorem is referenced by:  perfopn  23100