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Theorem elflim 17682
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 16680 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 451 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  J  e.  Top )
3 fvssunirn 5567 . . . . 5  |-  ( Fil `  X )  C_  U. ran  Fil
43sseli 3189 . . . 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 17562 . . . . 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 16681 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 451 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  X  =  U. J )
109pweqd 3643 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ~P X  =  ~P U. J
)
117, 10sseqtrd 3227 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  C_ 
~P U. J )
12 eqid 2296 . . . . 5  |-  U. J  =  U. J
1312elflim2 17675 . . . 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 2363 . . 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 1632    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   {csn 3653   U.cuni 3843   ran crn 4706   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648   neicnei 16850   Filcfil 17556    fLim cflim 17645
This theorem is referenced by:  flimss2  17683  flimss1  17684  neiflim  17685  flimopn  17686  hausflim  17692  flimclslem  17695  flfnei  17702  fclsfnflim  17738  plimfil  25661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-topon 16655  df-fbas 17536  df-fil 17557  df-flim 17650
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