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Theorem perfcls 17391
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 17390 . . . 4  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( limPt `  J
) `  ( ( cls `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )
)
32sseq2d 3344 . . 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 17356 . . . . . 6  |-  ( J  e.  Fre  ->  J  e.  Top )
51clslp 17174 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
64, 5sylan 458 . . . . 5  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
76sseq1d 3343 . . . 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 3485 . . . . 5  |-  ( S 
C_  ( ( limPt `  J ) `  S
)  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  =  ( (
limPt `  J ) `  S ) )
9 ssun2 3479 . . . . . 6  |-  ( (
limPt `  J ) `  S )  C_  ( S  u.  ( ( limPt `  J ) `  S ) )
10 eqss 3331 . . . . . 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 886 . . . . 5  |-  ( ( S  u.  ( (
limPt `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )  <->  ( S  u.  ( (
limPt `  J ) `  S ) )  C_  ( ( limPt `  J
) `  S )
)
128, 11bitri 241 . . . 4  |-  ( S 
C_  ( ( limPt `  J ) `  S
)  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  C_  ( ( limPt `  J ) `  S ) )
137, 12syl6bbr 255 . . 3  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  C_  ( ( limPt `  J ) `  S )  <->  S  C_  (
( limPt `  J ) `  S ) ) )
143, 13bitr2d 246 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( S  C_  (
( limPt `  J ) `  S )  <->  ( ( cls `  J ) `  S )  C_  (
( limPt `  J ) `  ( ( cls `  J
) `  S )
) ) )
15 eqid 2412 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
161, 15restperf 17210 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
174, 16sylan 458 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
181clsss3 17086 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
19 eqid 2412 . . . . 5  |-  ( Jt  ( ( cls `  J
) `  S )
)  =  ( Jt  ( ( cls `  J
) `  S )
)
201, 19restperf 17210 . . . 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 457 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( Jt  ( ( cls `  J ) `
 S ) )  e. Perf 
<->  ( ( cls `  J
) `  S )  C_  ( ( limPt `  J
) `  ( ( cls `  J ) `  S ) ) ) )
224, 21sylan 458 . 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 277 1  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  ( Jt  ( ( cls `  J ) `  S
) )  e. Perf )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    u. cun 3286    C_ wss 3288   U.cuni 3983   ` cfv 5421  (class class class)co 6048   ↾t crest 13611   Topctop 16921   clsccl 17045   limPtclp 17161  Perfcperf 17162   Frect1 17333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-oadd 6695  df-er 6872  df-en 7077  df-fin 7080  df-fi 7382  df-rest 13613  df-topgen 13630  df-top 16926  df-bases 16928  df-topon 16929  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-t1 17340
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