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

Theorem flimval 20976
Description: The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimval.1  |-  X  = 
U. J
Assertion
Ref Expression
flimval  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  ( J  fLim  F )  =  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) } )
Distinct variable groups:    x, F    x, J    x, X

Proof of Theorem flimval
Dummy variables  f 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimval.1 . . . . 5  |-  X  = 
U. J
21topopn 19934 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
32adantr 466 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  X  e.  J
)
4 rabexg 4574 . . 3  |-  ( X  e.  J  ->  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) }  e.  _V )
53, 4syl 17 . 2  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  { x  e.  X  |  ( ( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) }  e.  _V )
6 simpl 458 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  j  =  J )
76unieqd 4229 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
87, 1syl6eqr 2481 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
96fveq2d 5885 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( nei `  j
)  =  ( nei `  J ) )
109fveq1d 5883 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
11 simpr 462 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
1210, 11sseq12d 3493 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( nei `  j ) `  {
x } )  C_  f 
<->  ( ( nei `  J
) `  { x } )  C_  F
) )
138pweqd 3986 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P U. j  =  ~P X )
1411, 13sseq12d 3493 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f  C_  ~P U. j  <->  F  C_  ~P X
) )
1512, 14anbi12d 715 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( ( nei `  j ) `
 { x }
)  C_  f  /\  f  C_  ~P U. j
)  <->  ( ( ( nei `  J ) `
 { x }
)  C_  F  /\  F  C_  ~P X ) ) )
168, 15rabeqbidv 3075 . . 3  |-  ( ( j  =  J  /\  f  =  F )  ->  { x  e.  U. j  |  ( (
( nei `  j
) `  { x } )  C_  f  /\  f  C_  ~P U. j ) }  =  { x  e.  X  |  ( ( ( nei `  J ) `
 { x }
)  C_  F  /\  F  C_  ~P X ) } )
17 df-flim 20952 . . 3  |-  fLim  =  ( j  e.  Top ,  f  e.  U. ran  Fil  |->  { x  e.  U. j  |  ( (
( nei `  j
) `  { x } )  C_  f  /\  f  C_  ~P U. j ) } )
1816, 17ovmpt2ga 6440 . 2  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\ 
{ x  e.  X  |  ( ( ( nei `  J ) `
 { x }
)  C_  F  /\  F  C_  ~P X ) }  e.  _V )  ->  ( J  fLim  F
)  =  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) } )
195, 18mpd3an3 1361 1  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  ( J  fLim  F )  =  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   {crab 2775   _Vcvv 3080    C_ wss 3436   ~Pcpw 3981   {csn 3998   U.cuni 4219   ran crn 4854   ` cfv 5601  (class class class)co 6305   Topctop 19915   neicnei 20111   Filcfil 20858    fLim cflim 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-top 19919  df-flim 20952
This theorem is referenced by:  elflim2  20977
  Copyright terms: Public domain W3C validator