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Theorem filunirn 20146
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 5876 . . . . . 6  |-  ( fBas `  y )  e.  _V
21rabex 4598 . . . . 5  |-  { w  e.  ( fBas `  y
)  |  A. z  e.  ~P  y ( ( w  i^i  ~P z
)  =/=  (/)  ->  z  e.  w ) }  e.  _V
3 df-fil 20110 . . . . 5  |-  Fil  =  ( y  e.  _V  |->  { w  e.  ( fBas `  y )  | 
A. z  e.  ~P  y ( ( w  i^i  ~P z )  =/=  (/)  ->  z  e.  w ) } )
42, 3fnmpti 5709 . . . 4  |-  Fil  Fn  _V
5 fnunirn 6153 . . . 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 20145 . . . . . . 7  |-  ( F  e.  ( Fil `  x
)  ->  U. F  =  x )
87fveq2d 5870 . . . . . 6  |-  ( F  e.  ( Fil `  x
)  ->  ( Fil ` 
U. F )  =  ( Fil `  x
) )
98eleq2d 2537 . . . . 5  |-  ( F  e.  ( Fil `  x
)  ->  ( F  e.  ( Fil `  U. F )  <->  F  e.  ( Fil `  x ) ) )
109ibir 242 . . . 4  |-  ( F  e.  ( Fil `  x
)  ->  F  e.  ( Fil `  U. F
) )
1110rexlimivw 2952 . . 3  |-  ( E. x  e.  _V  F  e.  ( Fil `  x
)  ->  F  e.  ( Fil `  U. F
) )
126, 11sylbi 195 . 2  |-  ( F  e.  U. ran  Fil  ->  F  e.  ( Fil `  U. F ) )
13 fvssunirn 5889 . . 3  |-  ( Fil `  U. F )  C_  U.
ran  Fil
1413sseli 3500 . 2  |-  ( F  e.  ( Fil `  U. F )  ->  F  e.  U. ran  Fil )
1512, 14impbii 188 1  |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    i^i cin 3475   (/)c0 3785   ~Pcpw 4010   U.cuni 4245   ran crn 5000    Fn wfn 5583   ` cfv 5588   fBascfbas 18205   Filcfil 20109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-fbas 18215  df-fil 20110
This theorem is referenced by:  flimfil  20233  isfcls  20273
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