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Theorem filunirn 20828
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 5891 . . . . . 6  |-  ( fBas `  y )  e.  _V
21rabex 4576 . . . . 5  |-  { w  e.  ( fBas `  y
)  |  A. z  e.  ~P  y ( ( w  i^i  ~P z
)  =/=  (/)  ->  z  e.  w ) }  e.  _V
3 df-fil 20792 . . . . 5  |-  Fil  =  ( y  e.  _V  |->  { w  e.  ( fBas `  y )  | 
A. z  e.  ~P  y ( ( w  i^i  ~P z )  =/=  (/)  ->  z  e.  w ) } )
42, 3fnmpti 5724 . . . 4  |-  Fil  Fn  _V
5 fnunirn 6173 . . . 4  |-  ( Fil 
Fn  _V  ->  ( F  e.  U. ran  Fil  <->  E. x  e.  _V  F  e.  ( Fil `  x
) ) )
64, 5ax-mp 5 . . 3  |-  ( F  e.  U. ran  Fil  <->  E. x  e.  _V  F  e.  ( Fil `  x
) )
7 filunibas 20827 . . . . . . 7  |-  ( F  e.  ( Fil `  x
)  ->  U. F  =  x )
87fveq2d 5885 . . . . . 6  |-  ( F  e.  ( Fil `  x
)  ->  ( Fil ` 
U. F )  =  ( Fil `  x
) )
98eleq2d 2499 . . . . 5  |-  ( F  e.  ( Fil `  x
)  ->  ( F  e.  ( Fil `  U. F )  <->  F  e.  ( Fil `  x ) ) )
109ibir 245 . . . 4  |-  ( F  e.  ( Fil `  x
)  ->  F  e.  ( Fil `  U. F
) )
1110rexlimivw 2921 . . 3  |-  ( E. x  e.  _V  F  e.  ( Fil `  x
)  ->  F  e.  ( Fil `  U. F
) )
126, 11sylbi 198 . 2  |-  ( F  e.  U. ran  Fil  ->  F  e.  ( Fil `  U. F ) )
13 fvssunirn 5904 . . 3  |-  ( Fil `  U. F )  C_  U.
ran  Fil
1413sseli 3466 . 2  |-  ( F  e.  ( Fil `  U. F )  ->  F  e.  U. ran  Fil )
1512, 14impbii 190 1  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783   {crab 2786   _Vcvv 3087    i^i cin 3441   (/)c0 3767   ~Pcpw 3985   U.cuni 4222   ran crn 4855    Fn wfn 5596   ` cfv 5601   fBascfbas 18893   Filcfil 20791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-fbas 18902  df-fil 20792
This theorem is referenced by:  flimfil  20915  isfcls  20955
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