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Theorem filunirn 17836
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 5683 . . . . . 6  |-  ( fBas `  y )  e.  _V
21rabex 4296 . . . . 5  |-  { w  e.  ( fBas `  y
)  |  A. z  e.  ~P  y ( ( w  i^i  ~P z
)  =/=  (/)  ->  z  e.  w ) }  e.  _V
3 df-fil 17800 . . . . 5  |-  Fil  =  ( y  e.  _V  |->  { w  e.  ( fBas `  y )  | 
A. z  e.  ~P  y ( ( w  i^i  ~P z )  =/=  (/)  ->  z  e.  w ) } )
42, 3fnmpti 5514 . . . 4  |-  Fil  Fn  _V
5 fnunirn 5939 . . . 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 17835 . . . . . . 7  |-  ( F  e.  ( Fil `  x
)  ->  U. F  =  x )
87fveq2d 5673 . . . . . 6  |-  ( F  e.  ( Fil `  x
)  ->  ( Fil ` 
U. F )  =  ( Fil `  x
) )
98eleq2d 2455 . . . . 5  |-  ( F  e.  ( Fil `  x
)  ->  ( F  e.  ( Fil `  U. F )  <->  F  e.  ( Fil `  x ) ) )
109ibir 234 . . . 4  |-  ( F  e.  ( Fil `  x
)  ->  F  e.  ( Fil `  U. F
) )
1110rexlimivw 2770 . . 3  |-  ( E. x  e.  _V  F  e.  ( Fil `  x
)  ->  F  e.  ( Fil `  U. F
) )
126, 11sylbi 188 . 2  |-  ( F  e.  U. ran  Fil  ->  F  e.  ( Fil `  U. F ) )
13 fvssunirn 5695 . . 3  |-  ( Fil `  U. F )  C_  U.
ran  Fil
1413sseli 3288 . 2  |-  ( F  e.  ( Fil `  U. F )  ->  F  e.  U. ran  Fil )
1512, 14impbii 181 1  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   {crab 2654   _Vcvv 2900    i^i cin 3263   (/)c0 3572   ~Pcpw 3743   U.cuni 3958   ran crn 4820    Fn wfn 5390   ` cfv 5395   fBascfbas 16616   Filcfil 17799
This theorem is referenced by:  flimfil  17923  isfcls  17963
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403  df-fbas 16624  df-fil 17800
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