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

Theorem flimtop 20756
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
flimtop  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )

Proof of Theorem flimtop
StepHypRef Expression
1 eqid 2402 . . . 4  |-  U. J  =  U. J
21elflim2 20755 . . 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
) ) )
32simplbi 458 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  Top  /\  F  e. 
U. ran  Fil  /\  F  C_ 
~P U. J ) )
43simp1d 1009 1  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    e. wcel 1842    C_ wss 3413   ~Pcpw 3954   {csn 3971   U.cuni 4190   ran crn 4823   ` cfv 5568  (class class class)co 6277   Topctop 19684   neicnei 19889   Filcfil 20636    fLim cflim 20725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-top 19689  df-flim 20730
This theorem is referenced by:  flimfil  20760  flimtopon  20761  flimss1  20764  flimclsi  20769  hausflimlem  20770  flimsncls  20777  cnpflfi  20790  flimfcls  20817  flimfnfcls  20819
  Copyright terms: Public domain W3C validator