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Theorem elflim 17666
Description: The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
elflim  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )

Proof of Theorem elflim
StepHypRef Expression
1 topontop 16664 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 451 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  J  e.  Top )
3 fvssunirn 5551 . . . . 5  |-  ( Fil `  X )  C_  U. ran  Fil
43sseli 3176 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  U.
ran  Fil )
54adantl 452 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  e.  U. ran  Fil )
6 filsspw 17546 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ~P X )
76adantl 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  C_ 
~P X )
8 toponuni 16665 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 451 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  X  =  U. J )
109pweqd 3630 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ~P X  =  ~P U. J
)
117, 10sseqtrd 3214 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  C_ 
~P U. J )
12 eqid 2283 . . . . 5  |-  U. J  =  U. J
1312elflim2 17659 . . . 4  |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P U. J
)  /\  ( A  e.  U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
1413baib 871 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P U. J
)  ->  ( A  e.  ( J  fLim  F
)  <->  ( A  e. 
U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
152, 5, 11, 14syl3anc 1182 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
169eleq2d 2350 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  X  <->  A  e.  U. J ) )
1716anbi1d 685 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  F
)  <->  ( A  e. 
U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
1815, 17bitr4d 247 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   ran crn 4690   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632   neicnei 16834   Filcfil 17540    fLim cflim 17629
This theorem is referenced by:  flimss2  17667  flimss1  17668  neiflim  17669  flimopn  17670  hausflim  17676  flimclslem  17679  flfnei  17686  fclsfnflim  17722  plimfil  25558
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  ax-un 4512
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-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-topon 16639  df-fbas 17520  df-fil 17541  df-flim 17634
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