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

Theorem isperf 18760
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 5696 . . . 4  |-  ( j  =  J  ->  ( limPt `  j )  =  ( limPt `  J )
)
2 unieq 4104 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
3 lpfval.1 . . . . 5  |-  X  = 
U. J
42, 3syl6eqr 2493 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
51, 4fveq12d 5702 . . 3  |-  ( j  =  J  ->  (
( limPt `  j ) `  U. j )  =  ( ( limPt `  J
) `  X )
)
65, 4eqeq12d 2457 . 2  |-  ( j  =  J  ->  (
( ( limPt `  j
) `  U. j )  =  U. j  <->  ( ( limPt `  J ) `  X )  =  X ) )
7 df-perf 18746 . 2  |- Perf  =  {
j  e.  Top  | 
( ( limPt `  j
) `  U. j )  =  U. j }
86, 7elrab2 3124 1  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   U.cuni 4096   ` cfv 5423   Topctop 18503   limPtclp 18743  Perfcperf 18744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-perf 18746
This theorem is referenced by:  isperf2  18761  perflp  18763  perftop  18765  restperf  18793
  Copyright terms: Public domain W3C validator