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Theorem lpval 16798
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
StepHypRef Expression
1 lpfval.1 . . . . 5  |-  X  = 
U. J
21lpfval 16797 . . . 4  |-  ( J  e.  Top  ->  ( limPt `  J )  =  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) )
32fveq1d 5425 . . 3  |-  ( J  e.  Top  ->  (
( limPt `  J ) `  S )  =  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) `  S ) )
43adantr 453 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J
) `  ( y  \  { x } ) ) } ) `  S ) )
51topopn 16579 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4107 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 17 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 473 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
9 ssdifss 3249 . . . . . 6  |-  ( S 
C_  X  ->  ( S  \  { x }
)  C_  X )
101clsss3 16723 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( ( cls `  J ) `  ( S  \  { x } ) )  C_  X )
1110sseld 3121 . . . . . 6  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  X ) )
129, 11sylan2 462 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  X ) )
1312abssdv 3189 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  C_  X
)
145adantr 453 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  e.  J )
15 ssexg 4100 . . . 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 645 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  e.  _V )
17 difeq1 3229 . . . . . . 7  |-  ( y  =  S  ->  (
y  \  { x } )  =  ( S  \  { x } ) )
1817fveq2d 5427 . . . . . 6  |-  ( y  =  S  ->  (
( cls `  J
) `  ( y  \  { x } ) )  =  ( ( cls `  J ) `
 ( S  \  { x } ) ) )
1918eleq2d 2323 . . . . 5  |-  ( y  =  S  ->  (
x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) )  <->  x  e.  (
( cls `  J
) `  ( S  \  { x } ) ) ) )
2019abbidv 2370 . . . 4  |-  ( y  =  S  ->  { x  |  x  e.  (
( cls `  J
) `  ( y  \  { x } ) ) }  =  {
x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) } )
21 eqid 2256 . . . 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 5499 . . 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 645 . 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 2288 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2242   _Vcvv 2740    \ cdif 3091    C_ wss 3094   ~Pcpw 3566   {csn 3581   U.cuni 3768    e. cmpt 4017   ` cfv 4638   Topctop 16558   clsccl 16682   limPtclp 16793
This theorem is referenced by:  islp  16799  lpsscls  16800
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-top 16563  df-cld 16683  df-cls 16685  df-lp 16795
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