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Theorem isperf 19518
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 5872 . . . 4  |-  ( j  =  J  ->  ( limPt `  j )  =  ( limPt `  J )
)
2 unieq 4259 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
3 lpfval.1 . . . . 5  |-  X  = 
U. J
42, 3syl6eqr 2526 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
51, 4fveq12d 5878 . . 3  |-  ( j  =  J  ->  (
( limPt `  j ) `  U. j )  =  ( ( limPt `  J
) `  X )
)
65, 4eqeq12d 2489 . 2  |-  ( j  =  J  ->  (
( ( limPt `  j
) `  U. j )  =  U. j  <->  ( ( limPt `  J ) `  X )  =  X ) )
7 df-perf 19504 . 2  |- Perf  =  {
j  e.  Top  | 
( ( limPt `  j
) `  U. j )  =  U. j }
86, 7elrab2 3268 1  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   U.cuni 4251   ` cfv 5594   Topctop 19261   limPtclp 19501  Perfcperf 19502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-perf 19504
This theorem is referenced by:  isperf2  19519  perflp  19521  perftop  19523  restperf  19551
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