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Theorem restlp 17239
Description: The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
restcls.1  |-  X  = 
U. J
restcls.2  |-  K  =  ( Jt  Y )
Assertion
Ref Expression
restlp  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
( limPt `  K ) `  S )  =  ( ( ( limPt `  J
) `  S )  i^i  Y ) )

Proof of Theorem restlp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3 959 . . . . . . 7  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  S  C_  Y )
21ssdifssd 3477 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( S  \  { x }
)  C_  Y )
3 restcls.1 . . . . . . 7  |-  X  = 
U. J
4 restcls.2 . . . . . . 7  |-  K  =  ( Jt  Y )
53, 4restcls 17237 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  ( S  \  { x }
)  C_  Y )  ->  ( ( cls `  K
) `  ( S  \  { x } ) )  =  ( ( ( cls `  J
) `  ( S  \  { x } ) )  i^i  Y ) )
62, 5syld3an3 1229 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
( cls `  K
) `  ( S  \  { x } ) )  =  ( ( ( cls `  J
) `  ( S  \  { x } ) )  i^i  Y ) )
76eleq2d 2502 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( ( cls `  K ) `
 ( S  \  { x } ) )  <->  x  e.  (
( ( cls `  J
) `  ( S  \  { x } ) )  i^i  Y ) ) )
8 elin 3522 . . . 4  |-  ( x  e.  ( ( ( cls `  J ) `
 ( S  \  { x } ) )  i^i  Y )  <-> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  /\  x  e.  Y ) )
97, 8syl6bb 253 . . 3  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( ( cls `  K ) `
 ( S  \  { x } ) )  <->  ( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  /\  x  e.  Y ) ) )
10 simp1 957 . . . . . . . 8  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  J  e.  Top )
113toptopon 16990 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
1210, 11sylib 189 . . . . . . 7  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  J  e.  (TopOn `  X )
)
13 simp2 958 . . . . . . 7  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  Y  C_  X )
14 resttopon 17217 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  Y  C_  X )  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
1512, 13, 14syl2anc 643 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
164, 15syl5eqel 2519 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  K  e.  (TopOn `  Y )
)
17 topontop 16983 . . . . 5  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
1816, 17syl 16 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  K  e.  Top )
19 toponuni 16984 . . . . . 6  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
2016, 19syl 16 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  Y  =  U. K )
211, 20sseqtrd 3376 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  S  C_ 
U. K )
22 eqid 2435 . . . . 5  |-  U. K  =  U. K
2322islp 17196 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  U. K )  ->  ( x  e.  ( ( limPt `  K
) `  S )  <->  x  e.  ( ( cls `  K ) `  ( S  \  { x }
) ) ) )
2418, 21, 23syl2anc 643 . . 3  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( (
limPt `  K ) `  S )  <->  x  e.  ( ( cls `  K
) `  ( S  \  { x } ) ) ) )
25 elin 3522 . . . 4  |-  ( x  e.  ( ( (
limPt `  J ) `  S )  i^i  Y
)  <->  ( x  e.  ( ( limPt `  J
) `  S )  /\  x  e.  Y
) )
261, 13sstrd 3350 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  S  C_  X )
273islp 17196 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( limPt `  J ) `  S )  <->  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) ) )
2810, 26, 27syl2anc 643 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( (
limPt `  J ) `  S )  <->  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) ) )
2928anbi1d 686 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
( x  e.  ( ( limPt `  J ) `  S )  /\  x  e.  Y )  <->  ( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  /\  x  e.  Y ) ) )
3025, 29syl5bb 249 . . 3  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( ( ( limPt `  J ) `  S )  i^i  Y
)  <->  ( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  /\  x  e.  Y ) ) )
319, 24, 303bitr4d 277 . 2  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( (
limPt `  K ) `  S )  <->  x  e.  ( ( ( limPt `  J ) `  S
)  i^i  Y )
) )
3231eqrdv 2433 1  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
( limPt `  K ) `  S )  =  ( ( ( limPt `  J
) `  S )  i^i  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309    i^i cin 3311    C_ wss 3312   {csn 3806   U.cuni 4007   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   Topctop 16950  TopOnctopon 16951   clsccl 17074   limPtclp 17190
This theorem is referenced by:  restperf  17240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-oadd 6720  df-er 6897  df-en 7102  df-fin 7105  df-fi 7408  df-rest 13642  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-cld 17075  df-cls 17077  df-lp 17192
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