Theorem perflp 20958

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

Step	Hyp	Ref	Expression
1		lpfval.1	. . 3 |- X = U. J
2	1	isperf 20955	. 2 |- ( J e. Perf <-> ( J e. Top /\ ( ( limPt ` J ) ` X ) = X ) )
3	2	simprbi 480	1 |- ( J e. Perf -> ( ( limPt ` J ) ` X ) = X )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   U.cuni 4436   ` cfv 5888   Topctop 20698   limPtclp 20938  Perfcperf 20939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-perf 20941

This theorem is referenced by: (None)