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Theorem fgval 17565
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 17521 . . 3  |-  filGen  =  ( v  e.  _V , 
f  e.  ( fBas `  v )  |->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) } )
21a1i 10 . 2  |-  ( F  e.  ( fBas `  X
)  ->  filGen  =  ( v  e.  _V , 
f  e.  ( fBas `  v )  |->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) } ) )
3 pweq 3628 . . . . 5  |-  ( v  =  X  ->  ~P v  =  ~P X
)
43adantr 451 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ~P v  =  ~P X )
5 ineq1 3363 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
65neeq1d 2459 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
76adantl 452 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ( ( f  i^i 
~P x )  =/=  (/) 
<->  ( F  i^i  ~P x )  =/=  (/) ) )
84, 7rabeqbidv 2783 . . 3  |-  ( ( v  =  X  /\  f  =  F )  ->  { x  e.  ~P v  |  ( f  i^i  ~P x )  =/=  (/) }  =  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) } )
98adantl 452 . 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 5525 . . 3  |-  ( v  =  X  ->  ( fBas `  v )  =  ( fBas `  X
) )
1110adantl 452 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  v  =  X )  ->  ( fBas `  v )  =  ( fBas `  X
) )
12 elfvex 5555 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  _V )
13 id 19 . 2  |-  ( F  e.  ( fBas `  X
)  ->  F  e.  ( fBas `  X )
)
14 elfvdm 5554 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
15 pwexg 4194 . . 3  |-  ( X  e.  dom  fBas  ->  ~P X  e.  _V )
16 rabexg 4164 . . 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 5974 1  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   dom cdm 4689   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   fBascfbas 17518   filGencfg 17519
This theorem is referenced by:  elfg  17566  neifg  25732
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-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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-fg 17521
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