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Theorem filunirn 17577
Description: Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
filunirn  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )

Proof of Theorem filunirn
Dummy variables  y  w  z  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5539 . . . . . 6  |-  ( fBas `  y )  e.  _V
21rabex 4165 . . . . 5  |-  { w  e.  ( fBas `  y
)  |  A. z  e.  ~P  y ( ( w  i^i  ~P z
)  =/=  (/)  ->  z  e.  w ) }  e.  _V
3 df-fil 17541 . . . . 5  |-  Fil  =  ( y  e.  _V  |->  { w  e.  ( fBas `  y )  | 
A. z  e.  ~P  y ( ( w  i^i  ~P z )  =/=  (/)  ->  z  e.  w ) } )
42, 3fnmpti 5372 . . . 4  |-  Fil  Fn  _V
5 fnunirn 5778 . . . 4  |-  ( Fil 
Fn  _V  ->  ( F  e.  U. ran  Fil  <->  E. x  e.  _V  F  e.  ( Fil `  x
) ) )
64, 5ax-mp 8 . . 3  |-  ( F  e.  U. ran  Fil  <->  E. x  e.  _V  F  e.  ( Fil `  x
) )
7 filunibas 17576 . . . . . . 7  |-  ( F  e.  ( Fil `  x
)  ->  U. F  =  x )
87fveq2d 5529 . . . . . 6  |-  ( F  e.  ( Fil `  x
)  ->  ( Fil ` 
U. F )  =  ( Fil `  x
) )
98eleq2d 2350 . . . . 5  |-  ( F  e.  ( Fil `  x
)  ->  ( F  e.  ( Fil `  U. F )  <->  F  e.  ( Fil `  x ) ) )
109ibir 233 . . . 4  |-  ( F  e.  ( Fil `  x
)  ->  F  e.  ( Fil `  U. F
) )
1110rexlimivw 2663 . . 3  |-  ( E. x  e.  _V  F  e.  ( Fil `  x
)  ->  F  e.  ( Fil `  U. F
) )
126, 11sylbi 187 . 2  |-  ( F  e.  U. ran  Fil  ->  F  e.  ( Fil `  U. F ) )
13 fvssunirn 5551 . . 3  |-  ( Fil `  U. F )  C_  U.
ran  Fil
1413sseli 3176 . 2  |-  ( F  e.  ( Fil `  U. F )  ->  F  e.  U. ran  Fil )
1512, 14impbii 180 1  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   ran crn 4690    Fn wfn 5250   ` cfv 5255   fBascfbas 17518   Filcfil 17540
This theorem is referenced by:  flimfil  17664  isfcls  17704
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-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-fv 5263  df-fbas 17520  df-fil 17541
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