Theorem perflp 23433

Description: The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
lpfval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
perflp	⊢ (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)

Proof of Theorem perflp

Step	Hyp	Ref	Expression
1		lpfval.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	isperf 23430	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
3	2	simprbi 503	1 ⊢ (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∪ cuni 4866  ‘cfv 6527  Topctop 23172  limPtclp 23413  Perfcperf 23414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-perf 23416

This theorem is used by: (None)