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Theorem isfil 17644
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 17643 . 2  |-  Fil  =  ( z  e.  _V  |->  { f  e.  (
fBas `  z )  |  A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
2 pweq 3704 . . . 4  |-  ( z  =  X  ->  ~P z  =  ~P X
)
32adantr 451 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ~P z  =  ~P X )
4 ineq1 3439 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
54neeq1d 2534 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
6 eleq2 2419 . . . . 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 2824 . 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 5608 . 2  |-  ( z  =  X  ->  ( fBas `  z )  =  ( fBas `  X
) )
11 fvex 5622 . 2  |-  ( fBas `  z )  e.  _V
12 elfvdm 5637 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
131, 9, 10, 11, 12elmptrab2 17625 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 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   _Vcvv 2864    i^i cin 3227   (/)c0 3531   ~Pcpw 3701   dom cdm 4771   ` cfv 5337   fBascfbas 16471   Filcfil 17642
This theorem is referenced by:  filfbas  17645  filss  17650  isfil2  17653  ustfilxp  23518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fv 5345  df-fil 17643
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