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Theorem flimneiss 17990
Description: A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
flimneiss  |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A } )  C_  F )

Proof of Theorem flimneiss
StepHypRef Expression
1 eqid 2435 . . . 4  |-  U. J  =  U. J
21elflim2 17988 . . 3  |-  ( 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
) ) )
32simprbi 451 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( A  e.  U. J  /\  (
( nei `  J
) `  { A } )  C_  F
) )
43simprd 450 1  |-  ( A  e.  ( J  fLim  F )  ->  ( ( nei `  J ) `  { A } )  C_  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007   ran crn 4871   ` cfv 5446  (class class class)co 6073   Topctop 16950   neicnei 17153   Filcfil 17869    fLim cflim 17958
This theorem is referenced by:  flimnei  17991  flimfil  17993  flimss2  17996  flimss1  17997  flimcf  18006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-top 16955  df-flim 17963
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