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Theorem lpval 19446
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 19445 . . . 4  |-  ( J  e.  Top  ->  ( limPt `  J )  =  ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) )
32fveq1d 5868 . . 3  |-  ( J  e.  Top  ->  (
( limPt `  J ) `  S )  =  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) ) } ) `  S ) )
43adantr 465 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  ( ( y  e.  ~P X  |->  { x  |  x  e.  ( ( cls `  J
) `  ( y  \  { x } ) ) } ) `  S ) )
51topopn 19222 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4610 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 16 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 485 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
95adantr 465 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  e.  J )
10 ssdifss 3635 . . . . . 6  |-  ( S 
C_  X  ->  ( S  \  { x }
)  C_  X )
111clsss3 19366 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( ( cls `  J ) `  ( S  \  { x } ) )  C_  X )
1211sseld 3503 . . . . . 6  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X
)  ->  ( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  X ) )
1310, 12sylan2 474 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  X ) )
1413abssdv 3574 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  C_  X
)
159, 14ssexd 4594 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  e.  _V )
16 difeq1 3615 . . . . . . 7  |-  ( y  =  S  ->  (
y  \  { x } )  =  ( S  \  { x } ) )
1716fveq2d 5870 . . . . . 6  |-  ( y  =  S  ->  (
( cls `  J
) `  ( y  \  { x } ) )  =  ( ( cls `  J ) `
 ( S  \  { x } ) ) )
1817eleq2d 2537 . . . . 5  |-  ( y  =  S  ->  (
x  e.  ( ( cls `  J ) `
 ( y  \  { x } ) )  <->  x  e.  (
( cls `  J
) `  ( S  \  { x } ) ) ) )
1918abbidv 2603 . . . 4  |-  ( y  =  S  ->  { x  |  x  e.  (
( cls `  J
) `  ( y  \  { x } ) ) }  =  {
x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) } )
20 eqid 2467 . . . 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 5949 . . 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 661 . 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 2508 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113    \ cdif 3473    C_ wss 3476   ~Pcpw 4010   {csn 4027   U.cuni 4245    |-> cmpt 4505   ` cfv 5588   Topctop 19201   clsccl 19325   limPtclp 19441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-top 19206  df-cld 19326  df-cls 19328  df-lp 19443
This theorem is referenced by:  islp  19447  lpsscls  19448
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