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Theorem isperf 17215
Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
isperf  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X ) )

Proof of Theorem isperf
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . 4  |-  ( j  =  J  ->  ( limPt `  j )  =  ( limPt `  J )
)
2 unieq 4024 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
3 lpfval.1 . . . . 5  |-  X  = 
U. J
42, 3syl6eqr 2486 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
51, 4fveq12d 5734 . . 3  |-  ( j  =  J  ->  (
( limPt `  j ) `  U. j )  =  ( ( limPt `  J
) `  X )
)
65, 4eqeq12d 2450 . 2  |-  ( j  =  J  ->  (
( ( limPt `  j
) `  U. j )  =  U. j  <->  ( ( limPt `  J ) `  X )  =  X ) )
7 df-perf 17201 . 2  |- Perf  =  {
j  e.  Top  | 
( ( limPt `  j
) `  U. j )  =  U. j }
86, 7elrab2 3094 1  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   U.cuni 4015   ` cfv 5454   Topctop 16958   limPtclp 17198  Perfcperf 17199
This theorem is referenced by:  isperf2  17216  perflp  17218  perftop  17220  restperf  17248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-perf 17201
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