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Theorem restperf 19558
Description: Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
restcls.1  |-  X  = 
U. J
restcls.2  |-  K  =  ( Jt  Y )
Assertion
Ref Expression
restperf  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( K  e. Perf  <->  Y  C_  (
( limPt `  J ) `  Y ) ) )

Proof of Theorem restperf
StepHypRef Expression
1 restcls.2 . . . . 5  |-  K  =  ( Jt  Y )
2 restcls.1 . . . . . . 7  |-  X  = 
U. J
32toptopon 19307 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4 resttopon 19535 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  Y  C_  X )  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
53, 4sylanb 472 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( Jt  Y )  e.  (TopOn `  Y ) )
61, 5syl5eqel 2535 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X )  ->  K  e.  (TopOn `  Y
) )
7 topontop 19300 . . . 4  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
86, 7syl 16 . . 3  |-  ( ( J  e.  Top  /\  Y  C_  X )  ->  K  e.  Top )
9 eqid 2443 . . . . 5  |-  U. K  =  U. K
109isperf 19525 . . . 4  |-  ( K  e. Perf 
<->  ( K  e.  Top  /\  ( ( limPt `  K
) `  U. K )  =  U. K ) )
1110baib 903 . . 3  |-  ( K  e.  Top  ->  ( K  e. Perf  <->  ( ( limPt `  K ) `  U. K )  =  U. K ) )
128, 11syl 16 . 2  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( K  e. Perf  <->  ( ( limPt `  K ) `  U. K )  =  U. K ) )
13 dfss1 3688 . . 3  |-  ( Y 
C_  ( ( limPt `  J ) `  Y
)  <->  ( ( (
limPt `  J ) `  Y )  i^i  Y
)  =  Y )
14 ssid 3508 . . . . . 6  |-  Y  C_  Y
152, 1restlp 19557 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  Y  C_  Y )  ->  (
( limPt `  K ) `  Y )  =  ( ( ( limPt `  J
) `  Y )  i^i  Y ) )
1614, 15mp3an3 1314 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( ( limPt `  K
) `  Y )  =  ( ( (
limPt `  J ) `  Y )  i^i  Y
) )
17 toponuni 19301 . . . . . . 7  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
186, 17syl 16 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X )  ->  Y  =  U. K )
1918fveq2d 5860 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( ( limPt `  K
) `  Y )  =  ( ( limPt `  K ) `  U. K ) )
2016, 19eqtr3d 2486 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( ( ( limPt `  J ) `  Y
)  i^i  Y )  =  ( ( limPt `  K ) `  U. K ) )
2120, 18eqeq12d 2465 . . 3  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( ( ( (
limPt `  J ) `  Y )  i^i  Y
)  =  Y  <->  ( ( limPt `  K ) `  U. K )  =  U. K ) )
2213, 21syl5bb 257 . 2  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( Y  C_  (
( limPt `  J ) `  Y )  <->  ( ( limPt `  K ) `  U. K )  =  U. K ) )
2312, 22bitr4d 256 1  |-  ( ( J  e.  Top  /\  Y  C_  X )  -> 
( K  e. Perf  <->  Y  C_  (
( limPt `  J ) `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    i^i cin 3460    C_ wss 3461   U.cuni 4234   ` cfv 5578  (class class class)co 6281   ↾t crest 14695   Topctop 19267  TopOnctopon 19268   limPtclp 19508  Perfcperf 19509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-oadd 7136  df-er 7313  df-en 7519  df-fin 7522  df-fi 7873  df-rest 14697  df-topgen 14718  df-top 19272  df-bases 19274  df-topon 19275  df-cld 19393  df-cls 19395  df-lp 19510  df-perf 19511
This theorem is referenced by:  perfcls  19739  reperflem  21196  perfdvf  22180
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