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Theorem islp 17204
Description: The predicate " P is a limit point of  S." (Contributed by NM, 10-Feb-2007.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
islp  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )

Proof of Theorem islp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . 4  |-  X  = 
U. J
21lpval 17203 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) } )
32eleq2d 2503 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) } ) )
4 elex 2964 . . 3  |-  ( P  e.  ( ( cls `  J ) `  ( S  \  { P }
) )  ->  P  e.  _V )
5 id 20 . . . 4  |-  ( x  =  P  ->  x  =  P )
6 sneq 3825 . . . . . 6  |-  ( x  =  P  ->  { x }  =  { P } )
76difeq2d 3465 . . . . 5  |-  ( x  =  P  ->  ( S  \  { x }
)  =  ( S 
\  { P }
) )
87fveq2d 5732 . . . 4  |-  ( x  =  P  ->  (
( cls `  J
) `  ( S  \  { x } ) )  =  ( ( cls `  J ) `
 ( S  \  { P } ) ) )
95, 8eleq12d 2504 . . 3  |-  ( x  =  P  ->  (
x  e.  ( ( cls `  J ) `
 ( S  \  { x } ) )  <->  P  e.  (
( cls `  J
) `  ( S  \  { P } ) ) ) )
104, 9elab3 3089 . 2  |-  ( P  e.  { x  |  x  e.  ( ( cls `  J ) `
 ( S  \  { x } ) ) }  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) )
113, 10syl6bb 253 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( P  e.  ( ( limPt `  J ) `  S )  <->  P  e.  ( ( cls `  J
) `  ( S  \  { P } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422    \ cdif 3317    C_ wss 3320   {csn 3814   U.cuni 4015   ` cfv 5454   Topctop 16958   clsccl 17082   limPtclp 17198
This theorem is referenced by:  lpdifsn  17207  lpss3  17208  islp2  17209  islp3  17210  maxlp  17211  restlp  17247  lpcls  17428  limcnlp  19765  limcflflem  19767
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-top 16963  df-cld 17083  df-cls 17085  df-lp 17200
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