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Theorem elflim 20342
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 19297 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 465 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  J  e.  Top )
3 fvssunirn 5876 . . . . 5  |-  ( Fil `  X )  C_  U. ran  Fil
43sseli 3483 . . . 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 20222 . . . . 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 19298 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 465 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  X  =  U. J )
109pweqd 3999 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ~P X  =  ~P U. J
)
117, 10sseqtrd 3523 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  C_ 
~P U. J )
12 eqid 2441 . . . . 5  |-  U. J  =  U. J
1312elflim2 20335 . . . 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 1227 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
169eleq2d 2511 . . 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 972    = wceq 1381    e. wcel 1802    C_ wss 3459   ~Pcpw 3994   {csn 4011   U.cuni 4231   ran crn 4987   ` cfv 5575  (class class class)co 6278   Topctop 19264  TopOnctopon 19265   neicnei 19468   Filcfil 20216    fLim cflim 20305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fv 5583  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-fbas 18287  df-top 19269  df-topon 19272  df-fil 20217  df-flim 20310
This theorem is referenced by:  flimss2  20343  flimss1  20344  neiflim  20345  flimopn  20346  hausflim  20352  flimclslem  20355  flfnei  20362  fclsfnflim  20398
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