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Theorem isfil 20099
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 20098 . 2  |-  Fil  =  ( z  e.  _V  |->  { f  e.  (
fBas `  z )  |  A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
2 pweq 4013 . . . 4  |-  ( z  =  X  ->  ~P z  =  ~P X
)
32adantr 465 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ~P z  =  ~P X )
4 ineq1 3693 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
54neeq1d 2744 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
6 eleq2 2540 . . . . 5  |-  ( f  =  F  ->  (
x  e.  f  <->  x  e.  F ) )
75, 6imbi12d 320 . . . 4  |-  ( f  =  F  ->  (
( ( f  i^i 
~P x )  =/=  (/)  ->  x  e.  f )  <->  ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
87adantl 466 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ( ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f )  <->  ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
93, 8raleqbidv 3072 . 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 5865 . 2  |-  ( z  =  X  ->  ( fBas `  z )  =  ( fBas `  X
) )
11 fvex 5875 . 2  |-  ( fBas `  z )  e.  _V
12 elfvdm 5891 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
131, 9, 10, 11, 12elmptrab2 20080 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    i^i cin 3475   (/)c0 3785   ~Pcpw 4010   dom cdm 4999   ` cfv 5587   fBascfbas 18193   Filcfil 20097
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-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 5550  df-fun 5589  df-fv 5595  df-fil 20098
This theorem is referenced by:  filfbas  20100  filss  20105  isfil2  20108  ustfilxp  20466
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