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Theorem isfil 17840
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 17839 . 2  |-  Fil  =  ( z  e.  _V  |->  { f  e.  (
fBas `  z )  |  A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
2 pweq 3770 . . . 4  |-  ( z  =  X  ->  ~P z  =  ~P X
)
32adantr 452 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ~P z  =  ~P X )
4 ineq1 3503 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
54neeq1d 2588 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
6 eleq2 2473 . . . . 5  |-  ( f  =  F  ->  (
x  e.  f  <->  x  e.  F ) )
75, 6imbi12d 312 . . . 4  |-  ( f  =  F  ->  (
( ( f  i^i 
~P x )  =/=  (/)  ->  x  e.  f )  <->  ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
87adantl 453 . . 3  |-  ( ( z  =  X  /\  f  =  F )  ->  ( ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f )  <->  ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
93, 8raleqbidv 2884 . 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 5695 . 2  |-  ( z  =  X  ->  ( fBas `  z )  =  ( fBas `  X
) )
11 fvex 5709 . 2  |-  ( fBas `  z )  e.  _V
12 elfvdm 5724 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
131, 9, 10, 11, 12elmptrab2 17821 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   _Vcvv 2924    i^i cin 3287   (/)c0 3596   ~Pcpw 3767   dom cdm 4845   ` cfv 5421   fBascfbas 16652   Filcfil 17838
This theorem is referenced by:  filfbas  17841  filss  17846  isfil2  17849  ustfilxp  18203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fv 5429  df-fil 17839
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