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Theorem elflim2 17998
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 632 . 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 939 . . 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 678 . 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 17973 . . . 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 6290 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil )
)
6 flimval.1 . . . . . 6  |-  X  = 
U. J
76flimval 17997 . . . . 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 2505 . . . 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 3827 . . . . . . . . . 10  |-  ( x  =  A  ->  { x }  =  { A } )
109fveq2d 5734 . . . . . . . . 9  |-  ( x  =  A  ->  (
( nei `  J
) `  { x } )  =  ( ( nei `  J
) `  { A } ) )
1110sseq1d 3377 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( nei `  J
) `  { x } )  C_  F  <->  ( ( nei `  J
) `  { A } )  C_  F
) )
1211anbi1d 687 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( (
( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
) ) )
13 ancom 439 . . . . . . 7  |-  ( ( ( ( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
)  <->  ( F  C_  ~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
1412, 13syl6bb 254 . . . . . 6  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( F  C_ 
~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
1514elrab 3094 . . . . 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 774 . . . . 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 242 . . . 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 254 . . 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 625 . 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 271 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 set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   ~Pcpw 3801   {csn 3816   U.cuni 4017   ran crn 4881   ` cfv 5456  (class class class)co 6083   Topctop 16960   neicnei 17163   Filcfil 17879    fLim cflim 17968
This theorem is referenced by:  flimtop  17999  flimneiss  18000  flimelbas  18002  flimfil  18003  elflim  18005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-top 16965  df-flim 17973
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