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Theorem lpval 20092
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 20091 . . . 4  |-  ( J  e.  Top  ->  ( limPt `  J )  =  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) )
32fveq1d 5874 . . 3  |-  ( J  e.  Top  ->  (
( limPt `  J ) `  S )  =  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) `  S ) )
43adantr 466 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J
) `  ( y  \  { x } ) ) } ) `  S ) )
51topopn 19873 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4579 . . . . 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 487 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
95adantr 466 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  e.  J )
10 ssdifss 3593 . . . . . 6  |-  ( S 
C_  X  ->  ( S  \  { x }
)  C_  X )
111clsss3 20011 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( ( cls `  J ) `  ( S  \  { x } ) )  C_  X )
1211sseld 3460 . . . . . 6  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  X ) )
1310, 12sylan2 476 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  X ) )
1413abssdv 3532 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  C_  X
)
159, 14ssexd 4563 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  e.  _V )
16 difeq1 3573 . . . . . . 7  |-  ( y  =  S  ->  (
y  \  { x } )  =  ( S  \  { x } ) )
1716fveq2d 5876 . . . . . 6  |-  ( y  =  S  ->  (
( cls `  J
) `  ( y  \  { x } ) )  =  ( ( cls `  J ) `
 ( S  \  { x } ) ) )
1817eleq2d 2490 . . . . 5  |-  ( y  =  S  ->  (
x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) )  <->  x  e.  (
( cls `  J
) `  ( S  \  { x } ) ) ) )
1918abbidv 2556 . . . 4  |-  ( y  =  S  ->  { x  |  x  e.  (
( cls `  J
) `  ( y  \  { x } ) ) }  =  {
x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) } )
20 eqid 2420 . . . 4  |-  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J
) `  ( y  \  { x } ) ) } )  =  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } )
2119, 20fvmptg 5953 . . 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 } ) ) } )
228, 15, 21syl2anc 665 . 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 } ) ) } )
234, 22eqtrd 2461 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   {cab 2405   _Vcvv 3078    \ cdif 3430    C_ wss 3433   ~Pcpw 3976   {csn 3993   U.cuni 4213    |-> cmpt 4475   ` cfv 5592   Topctop 19854   clsccl 19970   limPtclp 20087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-top 19858  df-cld 19971  df-cls 19973  df-lp 20089
This theorem is referenced by:  islp  20093  lpsscls  20094
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