Theorem perftop 23135

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 23130	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘∪ 𝐽) = ∪ 𝐽))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ Perf → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∪ cuni 4851  ‘cfv 6494  Topctop 22872  limPtclp 23113  Perfcperf 23114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6450  df-fv 6502  df-perf 23116

This theorem is referenced by:  perfopn  23164