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

Theorem flimval 17952
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 16938 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
32adantr 452 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil )  ->  X  e.  J
)
4 rabexg 4317 . . 3  |-  ( X  e.  J  ->  { x  e.  X  |  (
( ( nei `  J
) `  { x } )  C_  F  /\  F  C_  ~P X
) }  e.  _V )
53, 4syl 16 . 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 444 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  j  =  J )
76unieqd 3990 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  U. J )
87, 1syl6eqr 2458 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  U. j  =  X )
96fveq2d 5695 . . . . . . 7  |-  ( ( j  =  J  /\  f  =  F )  ->  ( nei `  j
)  =  ( nei `  J ) )
109fveq1d 5693 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
11 simpr 448 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  f  =  F )
1210, 11sseq12d 3341 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( ( ( nei `  j ) `  {
x } )  C_  f 
<->  ( ( nei `  J
) `  { x } )  C_  F
) )
138pweqd 3768 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  ->  ~P U. j  =  ~P X )
1411, 13sseq12d 3341 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( f  C_  ~P U. j  <->  F  C_  ~P X
) )
1512, 14anbi12d 692 . . . 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 2915 . . 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 17928 . . 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 6166 . 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 1280 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2674   _Vcvv 2920    C_ wss 3284   ~Pcpw 3763   {csn 3778   U.cuni 3979   ran crn 4842   ` cfv 5417  (class class class)co 6044   Topctop 16917   neicnei 17120   Filcfil 17834    fLim cflim 17923
This theorem is referenced by:  elflim2  17953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-top 16922  df-flim 17928
  Copyright terms: Public domain W3C validator