MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elflim Structured version   Unicode version

Theorem elflim 20970
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 19925 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 466 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  J  e.  Top )
3 fvssunirn 5900 . . . . 5  |-  ( Fil `  X )  C_  U. ran  Fil
43sseli 3460 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  U.
ran  Fil )
54adantl 467 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  e.  U. ran  Fil )
6 filsspw 20850 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ~P X )
76adantl 467 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  C_ 
~P X )
8 toponuni 19926 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 466 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  X  =  U. J )
109pweqd 3984 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ~P X  =  ~P U. J
)
117, 10sseqtrd 3500 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  F  C_ 
~P U. J )
12 eqid 2422 . . . . 5  |-  U. J  =  U. J
1312elflim2 20963 . . . 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 911 . . 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 1264 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) ) )
169eleq2d 2492 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  X  <->  A  e.  U. J ) )
1716anbi1d 709 . 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 259 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    C_ wss 3436   ~Pcpw 3979   {csn 3996   U.cuni 4216   ran crn 4850   ` cfv 5597  (class class class)co 6301   Topctop 19901  TopOnctopon 19902   neicnei 20097   Filcfil 20844    fLim cflim 20933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-fbas 18952  df-top 19905  df-topon 19907  df-fil 20845  df-flim 20938
This theorem is referenced by:  flimss2  20971  flimss1  20972  neiflim  20973  flimopn  20974  hausflim  20980  flimclslem  20983  flfnei  20990  fclsfnflim  21026
  Copyright terms: Public domain W3C validator