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Theorem perfcls 17434
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 17433 . . . 4  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( limPt `  J
) `  ( ( cls `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )
)
32sseq2d 3378 . . 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 17399 . . . . . 6  |-  ( J  e.  Fre  ->  J  e.  Top )
51clslp 17217 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
64, 5sylan 459 . . . . 5  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
76sseq1d 3377 . . . 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 3519 . . . . 5  |-  ( S 
C_  ( ( limPt `  J ) `  S
)  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  =  ( (
limPt `  J ) `  S ) )
9 ssun2 3513 . . . . . 6  |-  ( (
limPt `  J ) `  S )  C_  ( S  u.  ( ( limPt `  J ) `  S ) )
10 eqss 3365 . . . . . 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 887 . . . . 5  |-  ( ( S  u.  ( (
limPt `  J ) `  S ) )  =  ( ( limPt `  J
) `  S )  <->  ( S  u.  ( (
limPt `  J ) `  S ) )  C_  ( ( limPt `  J
) `  S )
)
128, 11bitri 242 . . . 4  |-  ( S 
C_  ( ( limPt `  J ) `  S
)  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  C_  ( ( limPt `  J ) `  S ) )
137, 12syl6bbr 256 . . 3  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  C_  ( ( limPt `  J ) `  S )  <->  S  C_  (
( limPt `  J ) `  S ) ) )
143, 13bitr2d 247 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( S  C_  (
( limPt `  J ) `  S )  <->  ( ( cls `  J ) `  S )  C_  (
( limPt `  J ) `  ( ( cls `  J
) `  S )
) ) )
15 eqid 2438 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
161, 15restperf 17253 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
174, 16sylan 459 . 2  |-  ( ( J  e.  Fre  /\  S  C_  X )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
181clsss3 17128 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
19 eqid 2438 . . . . 5  |-  ( Jt  ( ( cls `  J
) `  S )
)  =  ( Jt  ( ( cls `  J
) `  S )
)
201, 19restperf 17253 . . . 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 458 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( Jt  ( ( cls `  J ) `
 S ) )  e. Perf 
<->  ( ( cls `  J
) `  S )  C_  ( ( limPt `  J
) `  ( ( cls `  J ) `  S ) ) ) )
224, 21sylan 459 . 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 278 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    u. cun 3320    C_ wss 3322   U.cuni 4017   ` cfv 5457  (class class class)co 6084   ↾t crest 13653   Topctop 16963   clsccl 17087   limPtclp 17203  Perfcperf 17204   Frect1 17376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-oadd 6731  df-er 6908  df-en 7113  df-fin 7116  df-fi 7419  df-rest 13655  df-topgen 13672  df-top 16968  df-bases 16970  df-topon 16971  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-t1 17383
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