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Theorem isfil 17542
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 17541 . 2  |-  Fil  =  ( z  e.  _V  |->  { f  e.  (
fBas `  z )  |  A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
2 pweq 3628 . . . 4  |-  ( z  =  X  ->  ~P z  =  ~P X
)
32adantr 451 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ~P z  =  ~P X )
4 ineq1 3363 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
54neeq1d 2459 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
6 eleq2 2344 . . . . 5  |-  ( f  =  F  ->  (
x  e.  f  <->  x  e.  F ) )
75, 6imbi12d 311 . . . 4  |-  ( f  =  F  ->  (
( ( f  i^i 
~P x )  =/=  (/)  ->  x  e.  f )  <->  ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
87adantl 452 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ( ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f )  <->  ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
93, 8raleqbidv 2748 . 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 5525 . 2  |-  ( z  =  X  ->  ( fBas `  z )  =  ( fBas `  X
) )
11 fvex 5539 . 2  |-  ( fBas `  z )  e.  _V
12 elfvdm 5554 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
131, 9, 10, 11, 12elmptrab2 17523 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   dom cdm 4689   ` cfv 5255   fBascfbas 17518   Filcfil 17540
This theorem is referenced by:  filfbas  17543  filss  17548  isfil2  17551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-fil 17541
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