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Theorem lpval 16834
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 16833 . . . 4  |-  ( J  e.  Top  ->  ( limPt `  J )  =  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) )
32fveq1d 5460 . . 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 16615 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4141 . . . . 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 3282 . . . . . 6  |-  ( S 
C_  X  ->  ( S  \  { x }
)  C_  X )
101clsss3 16759 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( ( cls `  J ) `  ( S  \  { x } ) )  C_  X )
1110sseld 3154 . . . . . 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 3222 . . . 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 4134 . . . 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 3262 . . . . . . 7  |-  ( y  =  S  ->  (
y  \  { x } )  =  ( S  \  { x } ) )
1817fveq2d 5462 . . . . . 6  |-  ( y  =  S  ->  (
( cls `  J
) `  ( y  \  { x } ) )  =  ( ( cls `  J ) `
 ( S  \  { x } ) ) )
1918eleq2d 2325 . . . . 5  |-  ( y  =  S  ->  (
x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) )  <->  x  e.  (
( cls `  J
) `  ( S  \  { x } ) ) ) )
2019abbidv 2372 . . . 4  |-  ( y  =  S  ->  { x  |  x  e.  (
( cls `  J
) `  ( y  \  { x } ) ) }  =  {
x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) } )
21 eqid 2258 . . . 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 5534 . . 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 2290 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 2244   _Vcvv 2763    \ cdif 3124    C_ wss 3127   ~Pcpw 3599   {csn 3614   U.cuni 3801    e. cmpt 4051   ` cfv 4673   Topctop 16594   clsccl 16718   limPtclp 16829
This theorem is referenced by:  islp  16835  lpsscls  16836
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-top 16599  df-cld 16719  df-cls 16721  df-lp 16831
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