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Theorem isperf 16882
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 5525 . . . 4  |-  ( j  =  J  ->  ( limPt `  j )  =  ( limPt `  J )
)
2 unieq 3836 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
3 lpfval.1 . . . . 5  |-  X  = 
U. J
42, 3syl6eqr 2333 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
51, 4fveq12d 5531 . . 3  |-  ( j  =  J  ->  (
( limPt `  j ) `  U. j )  =  ( ( limPt `  J
) `  X )
)
65, 4eqeq12d 2297 . 2  |-  ( j  =  J  ->  (
( ( limPt `  j
) `  U. j )  =  U. j  <->  ( ( limPt `  J ) `  X )  =  X ) )
7 df-perf 16869 . 2  |- Perf  =  {
j  e.  Top  | 
( ( limPt `  j
) `  U. j )  =  U. j }
86, 7elrab2 2925 1  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   U.cuni 3827   ` cfv 5255   Topctop 16631   limPtclp 16866  Perfcperf 16867
This theorem is referenced by:  isperf2  16883  perflp  16885  perftop  16887  restperf  16914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-perf 16869
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