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Theorem elflim 20207
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 19194 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 465 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  J  e.  Top )
3 fvssunirn 5887 . . . . 5  |-  ( Fil `  X )  C_  U. ran  Fil
43sseli 3500 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  U.
ran  Fil )
54adantl 466 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  e.  U. ran  Fil )
6 filsspw 20087 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ~P X )
76adantl 466 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  C_ 
~P X )
8 toponuni 19195 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 465 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  X  =  U. J )
109pweqd 4015 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ~P X  =  ~P U. J
)
117, 10sseqtrd 3540 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  C_ 
~P U. J )
12 eqid 2467 . . . . 5  |-  U. J  =  U. J
1312elflim2 20200 . . . 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 901 . . 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 1228 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
169eleq2d 2537 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  X  <->  A  e.  U. J ) )
1716anbi1d 704 . 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 256 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   ~Pcpw 4010   {csn 4027   U.cuni 4245   ran crn 5000   ` cfv 5586  (class class class)co 6282   Topctop 19161  TopOnctopon 19162   neicnei 19364   Filcfil 20081    fLim cflim 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18187  df-top 19166  df-topon 19169  df-fil 20082  df-flim 20175
This theorem is referenced by:  flimss2  20208  flimss1  20209  neiflim  20210  flimopn  20211  hausflim  20217  flimclslem  20220  flfnei  20227  fclsfnflim  20263
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