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Theorem restlp 17170
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 3429 . . . . . 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 17168 . . . . . 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 2455 . . . 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 3474 . . . 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 16922 . . . . . . . 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 17148 . . . . . . 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 2472 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  K  e.  (TopOn `  Y )
)
17 topontop 16915 . . . . 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 16916 . . . . . 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 3328 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  S  C_ 
U. K )
22 eqid 2388 . . . . 5  |-  U. K  =  U. K
2322islp 17128 . . . 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 3474 . . . 4  |-  ( x  e.  ( ( (
limPt `  J ) `  S )  i^i  Y
)  <->  ( x  e.  ( ( limPt `  J
) `  S )  /\  x  e.  Y
) )
261, 13sstrd 3302 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  S  C_  X )
273islp 17128 . . . . . 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 2386 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 1649    e. wcel 1717    \ cdif 3261    i^i cin 3263    C_ wss 3264   {csn 3758   U.cuni 3958   ` cfv 5395  (class class class)co 6021   ↾t crest 13576   Topctop 16882  TopOnctopon 16883   clsccl 17006   limPtclp 17122
This theorem is referenced by:  restperf  17171
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-oadd 6665  df-er 6842  df-en 7047  df-fin 7050  df-fi 7352  df-rest 13578  df-topgen 13595  df-top 16887  df-bases 16889  df-topon 16890  df-cld 17007  df-cls 17009  df-lp 17124
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