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Theorem filunirn 17593
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 5555 . . . . . 6  |-  ( fBas `  y )  e.  _V
21rabex 4181 . . . . 5  |-  { w  e.  ( fBas `  y
)  |  A. z  e.  ~P  y ( ( w  i^i  ~P z
)  =/=  (/)  ->  z  e.  w ) }  e.  _V
3 df-fil 17557 . . . . 5  |-  Fil  =  ( y  e.  _V  |->  { w  e.  ( fBas `  y )  | 
A. z  e.  ~P  y ( ( w  i^i  ~P z )  =/=  (/)  ->  z  e.  w ) } )
42, 3fnmpti 5388 . . . 4  |-  Fil  Fn  _V
5 fnunirn 5794 . . . 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 17592 . . . . . . 7  |-  ( F  e.  ( Fil `  x
)  ->  U. F  =  x )
87fveq2d 5545 . . . . . 6  |-  ( F  e.  ( Fil `  x
)  ->  ( Fil ` 
U. F )  =  ( Fil `  x
) )
98eleq2d 2363 . . . . 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 2676 . . 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 5567 . . 3  |-  ( Fil `  U. F )  C_  U.
ran  Fil
1413sseli 3189 . 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 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    i^i cin 3164   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   ran crn 4706    Fn wfn 5266   ` cfv 5271   fBascfbas 17534   Filcfil 17556
This theorem is referenced by:  flimfil  17680  isfcls  17720
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-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-fv 5279  df-fbas 17536  df-fil 17557
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