MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fgval Structured version   Unicode version

Theorem fgval 20103
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 18185 . . 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 4013 . . . . 5  |-  ( v  =  X  ->  ~P v  =  ~P X
)
43adantr 465 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ~P v  =  ~P X )
5 ineq1 3693 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
65neeq1d 2744 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
76adantl 466 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ( ( f  i^i 
~P x )  =/=  (/) 
<->  ( F  i^i  ~P x )  =/=  (/) ) )
84, 7rabeqbidv 3108 . . 3  |-  ( ( v  =  X  /\  f  =  F )  ->  { x  e.  ~P v  |  ( f  i^i  ~P x )  =/=  (/) }  =  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) } )
98adantl 466 . 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 5864 . . 3  |-  ( v  =  X  ->  ( fBas `  v )  =  ( fBas `  X
) )
1110adantl 466 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  v  =  X )  ->  ( fBas `  v )  =  ( fBas `  X
) )
12 elfvex 5891 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  _V )
13 id 22 . 2  |-  ( F  e.  ( fBas `  X
)  ->  F  e.  ( fBas `  X )
)
14 elfvdm 5890 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
15 pwexg 4631 . . 3  |-  ( X  e.  dom  fBas  ->  ~P X  e.  _V )
16 rabexg 4597 . . 3  |-  ( ~P X  e.  _V  ->  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) }  e.  _V )
1714, 15, 163syl 20 . 2  |-  ( F  e.  ( fBas `  X
)  ->  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) }  e.  _V )
182, 9, 11, 12, 13, 17ovmpt2dx 6411 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 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   _Vcvv 3113    i^i cin 3475   (/)c0 3785   ~Pcpw 4010   dom cdm 4999   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   fBascfbas 18174   filGencfg 18175
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-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-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fg 18185
This theorem is referenced by:  elfg  20104  restmetu  20822  neifg  29790
  Copyright terms: Public domain W3C validator