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Theorem perflp 22350
Description: The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
perflp (𝐽 ∈ Perf β†’ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋)

Proof of Theorem perflp
StepHypRef Expression
1 lpfval.1 . . 3 𝑋 = βˆͺ 𝐽
21isperf 22347 . 2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋))
32simprbi 498 1 (𝐽 ∈ Perf β†’ ((limPtβ€˜π½)β€˜π‘‹) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104  βˆͺ cuni 4844  β€˜cfv 6458  Topctop 22087  limPtclp 22330  Perfcperf 22331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-iota 6410  df-fv 6466  df-perf 22333
This theorem is referenced by: (None)
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