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Theorem perfcls 20036
Description: A subset of a perfect space is perfect iff its closure is perfect (and the closure is an actual perfect set, since it is both closed and perfect in the subspace topology). (Contributed by Mario Carneiro, 26-Dec-2016.)
Hypothesis
Ref Expression
lpcls.1  |-  X  = 
U. J
Assertion
Ref Expression
perfcls  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  ( Jt  ( ( cls `  J ) `  S
) )  e. Perf )
)

Proof of Theorem perfcls
StepHypRef Expression
1 lpcls.1 . . . . 5  |-  X  = 
U. J
21lpcls 20035 . . . 4  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( limPt `  J
) `  ( ( cls `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )
)
32sseq2d 3517 . . 3  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  C_  ( ( limPt `  J ) `  ( ( cls `  J
) `  S )
)  <->  ( ( cls `  J ) `  S
)  C_  ( ( limPt `  J ) `  S ) ) )
4 t1top 20001 . . . . . 6  |-  ( J  e.  Fre  ->  J  e.  Top )
51clslp 19819 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
64, 5sylan 469 . . . . 5  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
76sseq1d 3516 . . . 4  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  C_  ( ( limPt `  J ) `  S )  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  C_  ( ( limPt `  J ) `  S ) ) )
8 ssequn1 3660 . . . . 5  |-  ( S 
C_  ( ( limPt `  J ) `  S
)  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  =  ( (
limPt `  J ) `  S ) )
9 ssun2 3654 . . . . . 6  |-  ( (
limPt `  J ) `  S )  C_  ( S  u.  ( ( limPt `  J ) `  S ) )
10 eqss 3504 . . . . . 6  |-  ( ( S  u.  ( (
limPt `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )  <->  ( ( S  u.  (
( limPt `  J ) `  S ) )  C_  ( ( limPt `  J
) `  S )  /\  ( ( limPt `  J
) `  S )  C_  ( S  u.  (
( limPt `  J ) `  S ) ) ) )
119, 10mpbiran2 917 . . . . 5  |-  ( ( S  u.  ( (
limPt `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )  <->  ( S  u.  ( (
limPt `  J ) `  S ) )  C_  ( ( limPt `  J
) `  S )
)
128, 11bitri 249 . . . 4  |-  ( S 
C_  ( ( limPt `  J ) `  S
)  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  C_  ( ( limPt `  J ) `  S ) )
137, 12syl6bbr 263 . . 3  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  C_  ( ( limPt `  J ) `  S )  <->  S  C_  (
( limPt `  J ) `  S ) ) )
143, 13bitr2d 254 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( S  C_  (
( limPt `  J ) `  S )  <->  ( ( cls `  J ) `  S )  C_  (
( limPt `  J ) `  ( ( cls `  J
) `  S )
) ) )
15 eqid 2454 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
161, 15restperf 19855 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
174, 16sylan 469 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
181clsss3 19730 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
19 eqid 2454 . . . . 5  |-  ( Jt  ( ( cls `  J
) `  S )
)  =  ( Jt  ( ( cls `  J
) `  S )
)
201, 19restperf 19855 . . . 4  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  S )  C_  X )  ->  (
( Jt  ( ( cls `  J ) `  S
) )  e. Perf  <->  ( ( cls `  J ) `  S )  C_  (
( limPt `  J ) `  ( ( cls `  J
) `  S )
) ) )
2118, 20syldan 468 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( Jt  ( ( cls `  J ) `
 S ) )  e. Perf 
<->  ( ( cls `  J
) `  S )  C_  ( ( limPt `  J
) `  ( ( cls `  J ) `  S ) ) ) )
224, 21sylan 469 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( Jt  ( ( cls `  J ) `
 S ) )  e. Perf 
<->  ( ( cls `  J
) `  S )  C_  ( ( limPt `  J
) `  ( ( cls `  J ) `  S ) ) ) )
2314, 17, 223bitr4d 285 1  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  ( Jt  ( ( cls `  J ) `  S
) )  e. Perf )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    u. cun 3459    C_ wss 3461   U.cuni 4235   ` cfv 5570  (class class class)co 6270   ↾t crest 14913   Topctop 19564   clsccl 19689   limPtclp 19805  Perfcperf 19806   Frect1 19978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-oadd 7126  df-er 7303  df-en 7510  df-fin 7513  df-fi 7863  df-rest 14915  df-topgen 14936  df-top 19569  df-bases 19571  df-topon 19572  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-t1 19985
This theorem is referenced by: (None)
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