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Theorem isperf2 18889
Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
isperf2  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  X  C_  ( ( limPt `  J ) `  X ) ) )

Proof of Theorem isperf2
StepHypRef Expression
1 lpfval.1 . . 3  |-  X  = 
U. J
21isperf 18888 . 2  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X ) )
3 ssid 3484 . . . . 5  |-  X  C_  X
41lpss 18879 . . . . 5  |-  ( ( J  e.  Top  /\  X  C_  X )  -> 
( ( limPt `  J
) `  X )  C_  X )
53, 4mpan2 671 . . . 4  |-  ( J  e.  Top  ->  (
( limPt `  J ) `  X )  C_  X
)
6 eqss 3480 . . . . 5  |-  ( ( ( limPt `  J ) `  X )  =  X  <-> 
( ( ( limPt `  J ) `  X
)  C_  X  /\  X  C_  ( ( limPt `  J ) `  X
) ) )
76baib 896 . . . 4  |-  ( ( ( limPt `  J ) `  X )  C_  X  ->  ( ( ( limPt `  J ) `  X
)  =  X  <->  X  C_  (
( limPt `  J ) `  X ) ) )
85, 7syl 16 . . 3  |-  ( J  e.  Top  ->  (
( ( limPt `  J
) `  X )  =  X  <->  X  C_  ( (
limPt `  J ) `  X ) ) )
98pm5.32i 637 . 2  |-  ( ( J  e.  Top  /\  ( ( limPt `  J
) `  X )  =  X )  <->  ( J  e.  Top  /\  X  C_  ( ( limPt `  J
) `  X )
) )
102, 9bitri 249 1  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  X  C_  ( ( limPt `  J ) `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3437   U.cuni 4200   ` cfv 5527   Topctop 18631   limPtclp 18871  Perfcperf 18872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-top 18636  df-cld 18756  df-cls 18758  df-lp 18873  df-perf 18874
This theorem is referenced by:  isperf3  18890
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