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

Theorem isperf 19778
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 2516 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
51, 4fveq12d 5878 . . 3  |-  ( j  =  J  ->  (
( limPt `  j ) `  U. j )  =  ( ( limPt `  J
) `  X )
)
65, 4eqeq12d 2479 . 2  |-  ( j  =  J  ->  (
( ( limPt `  j
) `  U. j )  =  U. j  <->  ( ( limPt `  J ) `  X )  =  X ) )
7 df-perf 19764 . 2  |- Perf  =  {
j  e.  Top  | 
( ( limPt `  j
) `  U. j )  =  U. j }
86, 7elrab2 3259 1  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   U.cuni 4251   ` cfv 5594   Topctop 19520   limPtclp 19761  Perfcperf 19762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-perf 19764
This theorem is referenced by:  isperf2  19779  perflp  19781  perftop  19783  restperf  19811
  Copyright terms: Public domain W3C validator