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Theorem restlp 16913
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 difss 3303 . . . . . . 7  |-  ( S 
\  { x }
)  C_  S
2 simp3 957 . . . . . . 7  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  S  C_  Y )
31, 2syl5ss 3190 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( S  \  { x }
)  C_  Y )
4 restcls.1 . . . . . . 7  |-  X  = 
U. J
5 restcls.2 . . . . . . 7  |-  K  =  ( Jt  Y )
64, 5restcls 16911 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  ( S  \  { x }
)  C_  Y )  ->  ( ( cls `  K
) `  ( S  \  { x } ) )  =  ( ( ( cls `  J
) `  ( S  \  { x } ) )  i^i  Y ) )
73, 6syld3an3 1227 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
( cls `  K
) `  ( S  \  { x } ) )  =  ( ( ( cls `  J
) `  ( S  \  { x } ) )  i^i  Y ) )
87eleq2d 2350 . . . 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 ) ) )
9 elin 3358 . . . 4  |-  ( x  e.  ( ( ( cls `  J ) `
 ( S  \  { x } ) )  i^i  Y )  <-> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  /\  x  e.  Y ) )
108, 9syl6bb 252 . . 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 ) ) )
11 simp1 955 . . . . . . . 8  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  J  e.  Top )
124toptopon 16671 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
1311, 12sylib 188 . . . . . . 7  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  J  e.  (TopOn `  X )
)
14 simp2 956 . . . . . . 7  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  Y  C_  X )
15 resttopon 16892 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  Y  C_  X )  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
1613, 14, 15syl2anc 642 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( Jt  Y )  e.  (TopOn `  Y ) )
175, 16syl5eqel 2367 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  K  e.  (TopOn `  Y )
)
18 topontop 16664 . . . . 5  |-  ( K  e.  (TopOn `  Y
)  ->  K  e.  Top )
1917, 18syl 15 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  K  e.  Top )
20 toponuni 16665 . . . . . 6  |-  ( K  e.  (TopOn `  Y
)  ->  Y  =  U. K )
2117, 20syl 15 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  Y  =  U. K )
222, 21sseqtrd 3214 . . . 4  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  S  C_ 
U. K )
23 eqid 2283 . . . . 5  |-  U. K  =  U. K
2423islp 16872 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  U. K )  ->  ( x  e.  ( ( limPt `  K
) `  S )  <->  x  e.  ( ( cls `  K ) `  ( S  \  { x }
) ) ) )
2519, 22, 24syl2anc 642 . . 3  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( (
limPt `  K ) `  S )  <->  x  e.  ( ( cls `  K
) `  ( S  \  { x } ) ) ) )
26 elin 3358 . . . 4  |-  ( x  e.  ( ( (
limPt `  J ) `  S )  i^i  Y
)  <->  ( x  e.  ( ( limPt `  J
) `  S )  /\  x  e.  Y
) )
272, 14sstrd 3189 . . . . . 6  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  S  C_  X )
284islp 16872 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( limPt `  J ) `  S )  <->  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) ) )
2911, 27, 28syl2anc 642 . . . . 5  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( (
limPt `  J ) `  S )  <->  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) ) )
3029anbi1d 685 . . . 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 ) ) )
3126, 30syl5bb 248 . . 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 ) ) )
3210, 25, 313bitr4d 276 . 2  |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  (
x  e.  ( (
limPt `  K ) `  S )  <->  x  e.  ( ( ( limPt `  J ) `  S
)  i^i  Y )
) )
3332eqrdv 2281 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   U.cuni 3827   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631  TopOnctopon 16632   clsccl 16755   limPtclp 16866
This theorem is referenced by:  restperf  16914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-cls 16758  df-lp 16868
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