MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  perflp Structured version   Unicode version

Theorem perflp 19840
Description: The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
perflp  |-  ( J  e. Perf  ->  ( ( limPt `  J ) `  X
)  =  X )

Proof of Theorem perflp
StepHypRef Expression
1 lpfval.1 . . 3  |-  X  = 
U. J
21isperf 19837 . 2  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X ) )
32simprbi 462 1  |-  ( J  e. Perf  ->  ( ( limPt `  J ) `  X
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   U.cuni 4190   ` cfv 5525   Topctop 19578   limPtclp 19820  Perfcperf 19821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5489  df-fv 5533  df-perf 19823
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator