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Theorem flimval 20589
Description: The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimval.1  |-  X  = 
U. J
Assertion
Ref Expression
flimval  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  ( J  fLim  F )  =  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) } )
Distinct variable groups:    x, F    x, J    x, X

Proof of Theorem flimval
Dummy variables  f 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimval.1 . . . . 5  |-  X  = 
U. J
21topopn 19541 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
32adantr 465 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  X  e.  J
)
4 rabexg 4606 . . 3  |-  ( X  e.  J  ->  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) }  e.  _V )
53, 4syl 16 . 2  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  { x  e.  X  |  ( ( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) }  e.  _V )
6 simpl 457 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  j  =  J )
76unieqd 4261 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
87, 1syl6eqr 2516 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
96fveq2d 5876 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( nei `  j
)  =  ( nei `  J ) )
109fveq1d 5874 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
11 simpr 461 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
1210, 11sseq12d 3528 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( nei `  j ) `  {
x } )  C_  f 
<->  ( ( nei `  J
) `  { x } )  C_  F
) )
138pweqd 4020 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P U. j  =  ~P X )
1411, 13sseq12d 3528 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f  C_  ~P U. j  <->  F  C_  ~P X
) )
1512, 14anbi12d 710 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( ( nei `  j ) `
 { x }
)  C_  f  /\  f  C_  ~P U. j
)  <->  ( ( ( nei `  J ) `
 { x }
)  C_  F  /\  F  C_  ~P X ) ) )
168, 15rabeqbidv 3104 . . 3  |-  ( ( j  =  J  /\  f  =  F )  ->  { x  e.  U. j  |  ( (
( nei `  j
) `  { x } )  C_  f  /\  f  C_  ~P U. j ) }  =  { x  e.  X  |  ( ( ( nei `  J ) `
 { x }
)  C_  F  /\  F  C_  ~P X ) } )
17 df-flim 20565 . . 3  |-  fLim  =  ( j  e.  Top ,  f  e.  U. ran  Fil  |->  { x  e.  U. j  |  ( (
( nei `  j
) `  { x } )  C_  f  /\  f  C_  ~P U. j ) } )
1816, 17ovmpt2ga 6431 . 2  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\ 
{ x  e.  X  |  ( ( ( nei `  J ) `
 { x }
)  C_  F  /\  F  C_  ~P X ) }  e.  _V )  ->  ( J  fLim  F
)  =  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) } )
195, 18mpd3an3 1325 1  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  ( J  fLim  F )  =  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   {csn 4032   U.cuni 4251   ran crn 5009   ` cfv 5594  (class class class)co 6296   Topctop 19520   neicnei 19724   Filcfil 20471    fLim cflim 20560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-top 19525  df-flim 20565
This theorem is referenced by:  elflim2  20590
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