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Theorem lpfval 16886
Description: The limit point function on the subsets of a topology's base set. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
lpfval  |-  ( J  e.  Top  ->  ( limPt `  J )  =  ( x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `
 ( x  \  { y } ) ) } ) )
Distinct variable groups:    x, y, J    x, X, y

Proof of Theorem lpfval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . 4  |-  X  = 
U. J
21topopn 16668 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4210 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5761 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `  (
x  \  { y } ) ) } )  e.  _V )
52, 3, 43syl 18 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `  (
x  \  { y } ) ) } )  e.  _V )
6 unieq 3852 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2346 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3643 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 fveq2 5541 . . . . . . 7  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
109fveq1d 5543 . . . . . 6  |-  ( j  =  J  ->  (
( cls `  j
) `  ( x  \  { y } ) )  =  ( ( cls `  J ) `
 ( x  \  { y } ) ) )
1110eleq2d 2363 . . . . 5  |-  ( j  =  J  ->  (
y  e.  ( ( cls `  j ) `
 ( x  \  { y } ) )  <->  y  e.  ( ( cls `  J
) `  ( x  \  { y } ) ) ) )
1211abbidv 2410 . . . 4  |-  ( j  =  J  ->  { y  |  y  e.  ( ( cls `  j
) `  ( x  \  { y } ) ) }  =  {
y  |  y  e.  ( ( cls `  J
) `  ( x  \  { y } ) ) } )
138, 12mpteq12dv 4114 . . 3  |-  ( j  =  J  ->  (
x  e.  ~P U. j  |->  { y  |  y  e.  ( ( cls `  j ) `
 ( x  \  { y } ) ) } )  =  ( x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `
 ( x  \  { y } ) ) } ) )
14 df-lp 16884 . . 3  |-  limPt  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  { y  |  y  e.  ( ( cls `  j ) `
 ( x  \  { y } ) ) } ) )
1513, 14fvmptg 5616 . 2  |-  ( ( J  e.  Top  /\  ( x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `
 ( x  \  { y } ) ) } )  e. 
_V )  ->  ( limPt `  J )  =  ( x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `
 ( x  \  { y } ) ) } ) )
165, 15mpdan 649 1  |-  ( J  e.  Top  ->  ( limPt `  J )  =  ( x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `
 ( x  \  { y } ) ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    \ cdif 3162   ~Pcpw 3638   {csn 3653   U.cuni 3843    e. cmpt 4093   ` cfv 5271   Topctop 16647   clsccl 16771   limPtclp 16882
This theorem is referenced by:  lpval  16887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-lp 16884
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