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Theorem fgval 20663
Description: The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fgval  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
Distinct variable groups:    x, F    x, X

Proof of Theorem fgval
Dummy variables  v 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fg 18737 . . 3  |-  filGen  =  ( v  e.  _V , 
f  e.  ( fBas `  v )  |->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) } )
21a1i 11 . 2  |-  ( F  e.  ( fBas `  X
)  ->  filGen  =  ( v  e.  _V , 
f  e.  ( fBas `  v )  |->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) } ) )
3 pweq 3958 . . . . 5  |-  ( v  =  X  ->  ~P v  =  ~P X
)
43adantr 463 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ~P v  =  ~P X )
5 ineq1 3634 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
65neeq1d 2680 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
76adantl 464 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ( ( f  i^i 
~P x )  =/=  (/) 
<->  ( F  i^i  ~P x )  =/=  (/) ) )
84, 7rabeqbidv 3054 . . 3  |-  ( ( v  =  X  /\  f  =  F )  ->  { x  e.  ~P v  |  ( f  i^i  ~P x )  =/=  (/) }  =  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) } )
98adantl 464 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  (
v  =  X  /\  f  =  F )
)  ->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) }  =  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
10 fveq2 5849 . . 3  |-  ( v  =  X  ->  ( fBas `  v )  =  ( fBas `  X
) )
1110adantl 464 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  v  =  X )  ->  ( fBas `  v )  =  ( fBas `  X
) )
12 elfvex 5876 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  _V )
13 id 22 . 2  |-  ( F  e.  ( fBas `  X
)  ->  F  e.  ( fBas `  X )
)
14 elfvdm 5875 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
15 pwexg 4578 . . 3  |-  ( X  e.  dom  fBas  ->  ~P X  e.  _V )
16 rabexg 4544 . . 3  |-  ( ~P X  e.  _V  ->  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) }  e.  _V )
1714, 15, 163syl 18 . 2  |-  ( F  e.  ( fBas `  X
)  ->  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) }  e.  _V )
182, 9, 11, 12, 13, 17ovmpt2dx 6410 1  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2758   _Vcvv 3059    i^i cin 3413   (/)c0 3738   ~Pcpw 3955   dom cdm 4823   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   fBascfbas 18726   filGencfg 18727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-fg 18737
This theorem is referenced by:  elfg  20664  restmetu  21382  neifg  30599
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