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Theorem flimval 17871
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 16869 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
32adantr 451 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  X  e.  J
)
4 rabexg 4266 . . 3  |-  ( X  e.  J  ->  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) }  e.  _V )
53, 4syl 15 . 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 443 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  j  =  J )
76unieqd 3940 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
87, 1syl6eqr 2416 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
96fveq2d 5636 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( nei `  j
)  =  ( nei `  J ) )
109fveq1d 5634 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
11 simpr 447 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
1210, 11sseq12d 3293 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( nei `  j ) `  {
x } )  C_  f 
<->  ( ( nei `  J
) `  { x } )  C_  F
) )
138pweqd 3719 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P U. j  =  ~P X )
1411, 13sseq12d 3293 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f  C_  ~P U. j  <->  F  C_  ~P X
) )
1512, 14anbi12d 691 . . . 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 2868 . . 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 17847 . . 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 6103 . 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 1279 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   {crab 2632   _Vcvv 2873    C_ wss 3238   ~Pcpw 3714   {csn 3729   U.cuni 3929   ran crn 4793   ` cfv 5358  (class class class)co 5981   Topctop 16848   neicnei 17051   Filcfil 17753    fLim cflim 17842
This theorem is referenced by:  elflim2  17872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-top 16853  df-flim 17847
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