MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lpfval Unicode version

Theorem lpfval 17127
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 16904 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4326 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5906 . . 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 3968 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2439 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3749 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 fveq2 5670 . . . . . . 7  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
109fveq1d 5672 . . . . . 6  |-  ( j  =  J  ->  (
( cls `  j
) `  ( x  \  { y } ) )  =  ( ( cls `  J ) `
 ( x  \  { y } ) ) )
1110eleq2d 2456 . . . . 5  |-  ( j  =  J  ->  (
y  e.  ( ( cls `  j ) `
 ( x  \  { y } ) )  <->  y  e.  ( ( cls `  J
) `  ( x  \  { y } ) ) ) )
1211abbidv 2503 . . . 4  |-  ( j  =  J  ->  { y  |  y  e.  ( ( cls `  j
) `  ( x  \  { y } ) ) }  =  {
y  |  y  e.  ( ( cls `  J
) `  ( x  \  { y } ) ) } )
138, 12mpteq12dv 4230 . . 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 17125 . . 3  |-  limPt  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  { y  |  y  e.  ( ( cls `  j ) `
 ( x  \  { y } ) ) } ) )
1513, 14fvmptg 5745 . 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 1649    e. wcel 1717   {cab 2375   _Vcvv 2901    \ cdif 3262   ~Pcpw 3744   {csn 3759   U.cuni 3959    e. cmpt 4209   ` cfv 5396   Topctop 16883   clsccl 17007   limPtclp 17123
This theorem is referenced by:  lpval  17128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-top 16888  df-lp 17125
  Copyright terms: Public domain W3C validator