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Theorem filnet 26434
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 2296 . 2  |-  U_ n  e.  F  ( {
n }  X.  n
)  =  U_ n  e.  F  ( {
n }  X.  n
)
2 eqid 2296 . 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 26433 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 1531    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   {csn 3653   U_ciun 3921   {copab 4092    X. cxp 4703   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   DirRelcdir 14366   tailctail 14367   Filcfil 17556    FilMap cfm 17644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-dir 14368  df-tail 14369  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649
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