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Theorem elflim2 20979
Description: The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimval.1  |-  X  = 
U. J
Assertion
Ref Expression
elflim2  |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P X )  /\  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  F )
) )

Proof of Theorem elflim2
Dummy variables  x  f  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anass 655 . 2  |-  ( ( ( ( J  e. 
Top  /\  F  e.  U.
ran  Fil )  /\  F  C_ 
~P X )  /\  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  F
) )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  /\  ( F  C_  ~P X  /\  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  F )
) ) )
2 df-3an 987 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P X
)  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  /\  F  C_  ~P X
) )
32anbi1i 701 . 2  |-  ( ( ( J  e.  Top  /\  F  e.  U. ran  Fil 
/\  F  C_  ~P X )  /\  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  F
) )  <->  ( (
( J  e.  Top  /\  F  e.  U. ran  Fil )  /\  F  C_  ~P X )  /\  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  F
) ) )
4 df-flim 20954 . . . 4  |-  fLim  =  ( j  e.  Top ,  f  e.  U. ran  Fil  |->  { x  e.  U. j  |  ( (
( nei `  j
) `  { x } )  C_  f  /\  f  C_  ~P U. j ) } )
54elmpt2cl 6511 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil )
)
6 flimval.1 . . . . . 6  |-  X  = 
U. J
76flimval 20978 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  ( J  fLim  F )  =  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) } )
87eleq2d 2514 . . . 4  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  ( A  e.  ( J  fLim  F
)  <->  A  e.  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) } ) )
9 sneq 3978 . . . . . . . . . 10  |-  ( x  =  A  ->  { x }  =  { A } )
109fveq2d 5869 . . . . . . . . 9  |-  ( x  =  A  ->  (
( nei `  J
) `  { x } )  =  ( ( nei `  J
) `  { A } ) )
1110sseq1d 3459 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( nei `  J
) `  { x } )  C_  F  <->  ( ( nei `  J
) `  { A } )  C_  F
) )
1211anbi1d 711 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( (
( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
) ) )
13 ancom 452 . . . . . . 7  |-  ( ( ( ( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
)  <->  ( F  C_  ~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
1412, 13syl6bb 265 . . . . . 6  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( F  C_ 
~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
1514elrab 3196 . . . . 5  |-  ( A  e.  { x  e.  X  |  ( ( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) }  <->  ( A  e.  X  /\  ( F  C_  ~P X  /\  ( ( nei `  J
) `  { A } )  C_  F
) ) )
16 an12 806 . . . . 5  |-  ( ( A  e.  X  /\  ( F  C_  ~P X  /\  ( ( nei `  J
) `  { A } )  C_  F
) )  <->  ( F  C_ 
~P X  /\  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  F
) ) )
1715, 16bitri 253 . . . 4  |-  ( A  e.  { x  e.  X  |  ( ( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) }  <->  ( F  C_ 
~P X  /\  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  F
) ) )
188, 17syl6bb 265 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  ( A  e.  ( J  fLim  F
)  <->  ( F  C_  ~P X  /\  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  F
) ) ) )
195, 18biadan2 648 . 2  |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  /\  ( F  C_  ~P X  /\  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  F )
) ) )
201, 3, 193bitr4ri 282 1  |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P X )  /\  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  F )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   {crab 2741    C_ wss 3404   ~Pcpw 3951   {csn 3968   U.cuni 4198   ran crn 4835   ` cfv 5582  (class class class)co 6290   Topctop 19917   neicnei 20113   Filcfil 20860    fLim cflim 20949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-top 19921  df-flim 20954
This theorem is referenced by:  flimtop  20980  flimneiss  20981  flimelbas  20983  flimfil  20984  elflim  20986
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