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Theorem elflim2 20903
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 653 . 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 984 . . 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 699 . 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 20878 . . . 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 6516 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil )
)
6 flimval.1 . . . . . 6  |-  X  = 
U. J
76flimval 20902 . . . . 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 2490 . . . 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 4003 . . . . . . . . . 10  |-  ( x  =  A  ->  { x }  =  { A } )
109fveq2d 5876 . . . . . . . . 9  |-  ( x  =  A  ->  (
( nei `  J
) `  { x } )  =  ( ( nei `  J
) `  { A } ) )
1110sseq1d 3488 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( nei `  J
) `  { x } )  C_  F  <->  ( ( nei `  J
) `  { A } )  C_  F
) )
1211anbi1d 709 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( (
( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
) ) )
13 ancom 451 . . . . . . 7  |-  ( ( ( ( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
)  <->  ( F  C_  ~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
1412, 13syl6bb 264 . . . . . 6  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( F  C_ 
~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
1514elrab 3226 . . . . 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 804 . . . . 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 252 . . . 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 264 . . 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 646 . 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 281 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   {crab 2777    C_ wss 3433   ~Pcpw 3976   {csn 3993   U.cuni 4213   ran crn 4846   ` cfv 5592  (class class class)co 6296   Topctop 19841   neicnei 20037   Filcfil 20784    fLim cflim 20873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-top 19845  df-flim 20878
This theorem is referenced by:  flimtop  20904  flimneiss  20905  flimelbas  20907  flimfil  20908  elflim  20910
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