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Theorem perfcls 17199
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 17198 . . . 4  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( limPt `  J
) `  ( ( cls `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )
)
32sseq2d 3282 . . 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 17164 . . . . . 6  |-  ( J  e.  Fre  ->  J  e.  Top )
51clslp 16985 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
64, 5sylan 457 . . . . 5  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
76sseq1d 3281 . . . 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 3421 . . . . 5  |-  ( S 
C_  ( ( limPt `  J ) `  S
)  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  =  ( (
limPt `  J ) `  S ) )
9 ssun2 3415 . . . . . 6  |-  ( (
limPt `  J ) `  S )  C_  ( S  u.  ( ( limPt `  J ) `  S ) )
10 eqss 3270 . . . . . 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 885 . . . . 5  |-  ( ( S  u.  ( (
limPt `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )  <->  ( S  u.  ( (
limPt `  J ) `  S ) )  C_  ( ( limPt `  J
) `  S )
)
128, 11bitri 240 . . . 4  |-  ( S 
C_  ( ( limPt `  J ) `  S
)  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  C_  ( ( limPt `  J ) `  S ) )
137, 12syl6bbr 254 . . 3  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  C_  ( ( limPt `  J ) `  S )  <->  S  C_  (
( limPt `  J ) `  S ) ) )
143, 13bitr2d 245 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( S  C_  (
( limPt `  J ) `  S )  <->  ( ( cls `  J ) `  S )  C_  (
( limPt `  J ) `  ( ( cls `  J
) `  S )
) ) )
15 eqid 2358 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
161, 15restperf 17020 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
174, 16sylan 457 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
181clsss3 16902 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
19 eqid 2358 . . . . 5  |-  ( Jt  ( ( cls `  J
) `  S )
)  =  ( Jt  ( ( cls `  J
) `  S )
)
201, 19restperf 17020 . . . 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 456 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( Jt  ( ( cls `  J ) `
 S ) )  e. Perf 
<->  ( ( cls `  J
) `  S )  C_  ( ( limPt `  J
) `  ( ( cls `  J ) `  S ) ) ) )
224, 21sylan 457 . 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 276 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 176    /\ wa 358    = wceq 1642    e. wcel 1710    u. cun 3226    C_ wss 3228   U.cuni 3908   ` cfv 5337  (class class class)co 5945   ↾t crest 13424   Topctop 16737   clsccl 16861   limPtclp 16972  Perfcperf 16973   Frect1 17141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-oadd 6570  df-er 6747  df-en 6952  df-fin 6955  df-fi 7255  df-rest 13426  df-topgen 13443  df-top 16742  df-bases 16744  df-topon 16745  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-t1 17148
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