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Theorem filnet 26413
Description: A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Assertion
Ref Expression
filnet  |-  ( F  e.  ( Fil `  X
)  ->  E. d  e.  DirRel  E. f ( f : dom  d --> X  /\  F  =  ( ( X  FilMap  f ) `
 ran  ( tail `  d ) ) ) )
Distinct variable groups:    f, d, F    X, d, f

Proof of Theorem filnet
Dummy variables  x  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . 2  |-  U_ n  e.  F  ( {
n }  X.  n
)  =  U_ n  e.  F  ( {
n }  X.  n
)
2 eqid 2438 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  U_ n  e.  F  ( { n }  X.  n )  /\  y  e.  U_ n  e.  F  ( { n }  X.  n ) )  /\  ( 1st `  y ) 
C_  ( 1st `  x
) ) }  =  { <. x ,  y
>.  |  ( (
x  e.  U_ n  e.  F  ( {
n }  X.  n
)  /\  y  e.  U_ n  e.  F  ( { n }  X.  n ) )  /\  ( 1st `  y ) 
C_  ( 1st `  x
) ) }
31, 2filnetlem4 26412 1  |-  ( F  e.  ( Fil `  X
)  ->  E. d  e.  DirRel  E. f ( f : dom  d --> X  /\  F  =  ( ( X  FilMap  f ) `
 ran  ( tail `  d ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   {csn 3816   U_ciun 4095   {copab 4267    X. cxp 4878   dom cdm 4880   ran crn 4881   -->wf 5452   ` cfv 5456  (class class class)co 6083   1stc1st 6349   DirRelcdir 14675   tailctail 14676   Filcfil 17879    FilMap cfm 17967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-dir 14677  df-tail 14678  df-fbas 16701  df-fg 16702  df-fil 17880  df-fm 17972
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