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Theorem flimval 19548
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 18531 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
32adantr 465 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  X  e.  J
)
4 rabexg 4454 . . 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 4113 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
87, 1syl6eqr 2493 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
96fveq2d 5707 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( nei `  j
)  =  ( nei `  J ) )
109fveq1d 5705 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
11 simpr 461 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
1210, 11sseq12d 3397 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( nei `  j ) `  {
x } )  C_  f 
<->  ( ( nei `  J
) `  { x } )  C_  F
) )
138pweqd 3877 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P U. j  =  ~P X )
1411, 13sseq12d 3397 . . . . 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 2979 . . 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 19524 . . 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 6232 . 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 1315 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 1369    e. wcel 1756   {crab 2731   _Vcvv 2984    C_ wss 3340   ~Pcpw 3872   {csn 3889   U.cuni 4103   ran crn 4853   ` cfv 5430  (class class class)co 6103   Topctop 18510   neicnei 18713   Filcfil 19430    fLim cflim 19519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-top 18515  df-flim 19524
This theorem is referenced by:  elflim2  19549
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