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Theorem filnet 26331
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 2283 . 2  |-  U_ n  e.  F  ( {
n }  X.  n
)  =  U_ n  e.  F  ( {
n }  X.  n
)
2 eqid 2283 . 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 26330 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 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   {csn 3640   U_ciun 3905   {copab 4076    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   DirRelcdir 14350   tailctail 14351   Filcfil 17540    FilMap cfm 17628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-dir 14352  df-tail 14353  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633
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