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

Theorem flimelbas 20975
Description: A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
flimuni.1  |-  X  = 
U. J
Assertion
Ref Expression
flimelbas  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )

Proof of Theorem flimelbas
StepHypRef Expression
1 flimuni.1 . . . 4  |-  X  = 
U. J
21elflim2 20971 . . 3  |-  ( A  e.  ( J  fLim  F )  <->  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  F  C_  ~P X )  /\  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  F )
) )
32simprbi 466 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( A  e.  X  /\  (
( nei `  J
) `  { A } )  C_  F
) )
43simpld 461 1  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    C_ wss 3437   ~Pcpw 3980   {csn 3997   U.cuni 4217   ran crn 4852   ` cfv 5599  (class class class)co 6303   Topctop 19909   neicnei 20105   Filcfil 20852    fLim cflim 20941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-top 19913  df-flim 20946
This theorem is referenced by:  flimfil  20976  flimss2  20979  flimss1  20980  flimclsi  20985  hausflimi  20987  flimsncls  20993  cnpflfi  21006  cnflf  21009  cnflf2  21010  flimcfil  22275
  Copyright terms: Public domain W3C validator