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Theorem lpval 16867
Description: The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
lpval  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) } )
Distinct variable groups:    x, J    x, S    x, X

Proof of Theorem lpval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . . 5  |-  X  = 
U. J
21lpfval 16866 . . . 4  |-  ( J  e.  Top  ->  ( limPt `  J )  =  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) )
32fveq1d 5488 . . 3  |-  ( J  e.  Top  ->  (
( limPt `  J ) `  S )  =  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) `  S ) )
43adantr 451 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J
) `  ( y  \  { x } ) ) } ) `  S ) )
51topopn 16648 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4168 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 15 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 471 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
9 ssdifss 3308 . . . . . 6  |-  ( S 
C_  X  ->  ( S  \  { x }
)  C_  X )
101clsss3 16792 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( ( cls `  J ) `  ( S  \  { x } ) )  C_  X )
1110sseld 3180 . . . . . 6  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  X ) )
129, 11sylan2 460 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  X ) )
1312abssdv 3248 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  C_  X
)
145adantr 451 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  e.  J )
15 ssexg 4161 . . . 4  |-  ( ( { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) }  C_  X  /\  X  e.  J
)  ->  { x  |  x  e.  (
( cls `  J
) `  ( S  \  { x } ) ) }  e.  _V )
1613, 14, 15syl2anc 642 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  e.  _V )
17 difeq1 3288 . . . . . . 7  |-  ( y  =  S  ->  (
y  \  { x } )  =  ( S  \  { x } ) )
1817fveq2d 5490 . . . . . 6  |-  ( y  =  S  ->  (
( cls `  J
) `  ( y  \  { x } ) )  =  ( ( cls `  J ) `
 ( S  \  { x } ) ) )
1918eleq2d 2351 . . . . 5  |-  ( y  =  S  ->  (
x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) )  <->  x  e.  (
( cls `  J
) `  ( S  \  { x } ) ) ) )
2019abbidv 2398 . . . 4  |-  ( y  =  S  ->  { x  |  x  e.  (
( cls `  J
) `  ( y  \  { x } ) ) }  =  {
x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) } )
21 eqid 2284 . . . 4  |-  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J
) `  ( y  \  { x } ) ) } )  =  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } )
2220, 21fvmptg 5562 . . 3  |-  ( ( S  e.  ~P X  /\  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) }  e.  _V )  ->  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `  (
y  \  { x } ) ) } ) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `
 ( S  \  { x } ) ) } )
238, 16, 22syl2anc 642 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( y  e. 
~P X  |->  { x  |  x  e.  (
( cls `  J
) `  ( y  \  { x } ) ) } ) `  S )  =  {
x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) } )
244, 23eqtrd 2316 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   {cab 2270   _Vcvv 2789    \ cdif 3150    C_ wss 3153   ~Pcpw 3626   {csn 3641   U.cuni 3828    e. cmpt 4078   ` cfv 5221   Topctop 16627   clsccl 16751   limPtclp 16862
This theorem is referenced by:  islp  16868  lpsscls  16869
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-top 16632  df-cld 16752  df-cls 16754  df-lp 16864
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