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Theorem lpfval 17194
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 16971 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4375 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5957 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `  (
x  \  { y } ) ) } )  e.  _V )
52, 3, 43syl 19 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `  (
x  \  { y } ) ) } )  e.  _V )
6 unieq 4016 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2485 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3796 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 fveq2 5720 . . . . . . 7  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
109fveq1d 5722 . . . . . 6  |-  ( j  =  J  ->  (
( cls `  j
) `  ( x  \  { y } ) )  =  ( ( cls `  J ) `
 ( x  \  { y } ) ) )
1110eleq2d 2502 . . . . 5  |-  ( j  =  J  ->  (
y  e.  ( ( cls `  j ) `
 ( x  \  { y } ) )  <->  y  e.  ( ( cls `  J
) `  ( x  \  { y } ) ) ) )
1211abbidv 2549 . . . 4  |-  ( j  =  J  ->  { y  |  y  e.  ( ( cls `  j
) `  ( x  \  { y } ) ) }  =  {
y  |  y  e.  ( ( cls `  J
) `  ( x  \  { y } ) ) } )
138, 12mpteq12dv 4279 . . 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 17192 . . 3  |-  limPt  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  { y  |  y  e.  ( ( cls `  j ) `
 ( x  \  { y } ) ) } ) )
1513, 14fvmptg 5796 . 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 650 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 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    \ cdif 3309   ~Pcpw 3791   {csn 3806   U.cuni 4007    e. cmpt 4258   ` cfv 5446   Topctop 16950   clsccl 17074   limPtclp 17190
This theorem is referenced by:  lpval  17195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16955  df-lp 17192
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