Theorem perflp 23322

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 23319	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))
3	2	simprbi 502	1 ⊢ (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ∪ cuni 4871  ‘cfv 6536  Topctop 23061  limPtclp 23302  Perfcperf 23303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-perf 23305

This theorem is used by: (None)