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Theorem elflim2 20333
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 649 . 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 975 . . 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 695 . 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 20308 . . . 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 6512 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil )
)
6 flimval.1 . . . . . 6  |-  X  = 
U. J
76flimval 20332 . . . . 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 2537 . . . 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 4043 . . . . . . . . . 10  |-  ( x  =  A  ->  { x }  =  { A } )
109fveq2d 5876 . . . . . . . . 9  |-  ( x  =  A  ->  (
( nei `  J
) `  { x } )  =  ( ( nei `  J
) `  { A } ) )
1110sseq1d 3536 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( nei `  J
) `  { x } )  C_  F  <->  ( ( nei `  J
) `  { A } )  C_  F
) )
1211anbi1d 704 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( (
( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
) ) )
13 ancom 450 . . . . . . 7  |-  ( ( ( ( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
)  <->  ( F  C_  ~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
1412, 13syl6bb 261 . . . . . 6  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( F  C_ 
~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
1514elrab 3266 . . . . 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 795 . . . . 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 249 . . . 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 261 . . 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 642 . 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 278 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2821    C_ wss 3481   ~Pcpw 4016   {csn 4033   U.cuni 4251   ran crn 5006   ` cfv 5594  (class class class)co 6295   Topctop 19263   neicnei 19466   Filcfil 20214    fLim cflim 20303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-top 19268  df-flim 20308
This theorem is referenced by:  flimtop  20334  flimneiss  20335  flimelbas  20337  flimfil  20338  elflim  20340
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