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Theorem elflim2 17659
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 630 . 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 936 . . 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 676 . 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 17634 . . . 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 6061 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil )
)
6 flimval.1 . . . . . 6  |-  X  = 
U. J
76flimval 17658 . . . . 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 2350 . . . 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 3651 . . . . . . . . . 10  |-  ( x  =  A  ->  { x }  =  { A } )
109fveq2d 5529 . . . . . . . . 9  |-  ( x  =  A  ->  (
( nei `  J
) `  { x } )  =  ( ( nei `  J
) `  { A } ) )
1110sseq1d 3205 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( nei `  J
) `  { x } )  C_  F  <->  ( ( nei `  J
) `  { A } )  C_  F
) )
1211anbi1d 685 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( (
( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
) ) )
13 ancom 437 . . . . . . 7  |-  ( ( ( ( nei `  J
) `  { A } )  C_  F  /\  F  C_  ~P X
)  <->  ( F  C_  ~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
1412, 13syl6bb 252 . . . . . 6  |-  ( x  =  A  ->  (
( ( ( nei `  J ) `  {
x } )  C_  F  /\  F  C_  ~P X )  <->  ( F  C_ 
~P X  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
1514elrab 2923 . . . . 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 772 . . . . 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 240 . . . 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 252 . . 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 623 . 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 269 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   ran crn 4690   ` cfv 5255  (class class class)co 5858   Topctop 16631   neicnei 16834   Filcfil 17540    fLim cflim 17629
This theorem is referenced by:  flimtop  17660  flimneiss  17661  flimelbas  17663  flimfil  17664  elflim  17666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-flim 17634
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