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Theorem isfil 17884
Description: The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
isfil  |-  ( F  e.  ( Fil `  X
)  <->  ( F  e.  ( fBas `  X
)  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) ) )
Distinct variable groups:    x, F    x, X

Proof of Theorem isfil
Dummy variables  f 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fil 17883 . 2  |-  Fil  =  ( z  e.  _V  |->  { f  e.  (
fBas `  z )  |  A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
2 pweq 3804 . . . 4  |-  ( z  =  X  ->  ~P z  =  ~P X
)
32adantr 453 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ~P z  =  ~P X )
4 ineq1 3537 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
54neeq1d 2616 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
6 eleq2 2499 . . . . 5  |-  ( f  =  F  ->  (
x  e.  f  <->  x  e.  F ) )
75, 6imbi12d 313 . . . 4  |-  ( f  =  F  ->  (
( ( f  i^i 
~P x )  =/=  (/)  ->  x  e.  f )  <->  ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
87adantl 454 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ( ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f )  <->  ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
93, 8raleqbidv 2918 . 2  |-  ( ( z  =  X  /\  f  =  F )  ->  ( A. x  e. 
~P  z ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f )  <->  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) ) )
10 fveq2 5731 . 2  |-  ( z  =  X  ->  ( fBas `  z )  =  ( fBas `  X
) )
11 fvex 5745 . 2  |-  ( fBas `  z )  e.  _V
12 elfvdm 5760 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
131, 9, 10, 11, 12elmptrab2 17865 1  |-  ( F  e.  ( Fil `  X
)  <->  ( F  e.  ( fBas `  X
)  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958    i^i cin 3321   (/)c0 3630   ~Pcpw 3801   dom cdm 4881   ` cfv 5457   fBascfbas 16694   Filcfil 17882
This theorem is referenced by:  filfbas  17885  filss  17890  isfil2  17893  ustfilxp  18247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fv 5465  df-fil 17883
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