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Definition df-fbas 17515
Description: Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
Assertion
Ref Expression
df-fbas  |-  fBas  =  ( w  e.  _V  |->  { x  e.  ~P ~P w  |  (
x  =/=  (/)  /\  (/)  e/  x  /\  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P (
y  i^i  z )
)  =/=  (/) ) } )
Distinct variable group:    x, w, y, z

Detailed syntax breakdown of Definition df-fbas
StepHypRef Expression
1 cfbas 17513 . 2  class  fBas
2 vw . . 3  set  w
3 cvv 2790 . . 3  class  _V
4 vx . . . . . . 7  set  x
54cv 1623 . . . . . 6  class  x
6 c0 3457 . . . . . 6  class  (/)
75, 6wne 2448 . . . . 5  wff  x  =/=  (/)
86, 5wnel 2449 . . . . 5  wff  (/)  e/  x
9 vy . . . . . . . . . . . 12  set  y
109cv 1623 . . . . . . . . . . 11  class  y
11 vz . . . . . . . . . . . 12  set  z
1211cv 1623 . . . . . . . . . . 11  class  z
1310, 12cin 3153 . . . . . . . . . 10  class  ( y  i^i  z )
1413cpw 3627 . . . . . . . . 9  class  ~P (
y  i^i  z )
155, 14cin 3153 . . . . . . . 8  class  ( x  i^i  ~P ( y  i^i  z ) )
1615, 6wne 2448 . . . . . . 7  wff  ( x  i^i  ~P ( y  i^i  z ) )  =/=  (/)
1716, 11, 5wral 2545 . . . . . 6  wff  A. z  e.  x  ( x  i^i  ~P ( y  i^i  z ) )  =/=  (/)
1817, 9, 5wral 2545 . . . . 5  wff  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P ( y  i^i  z ) )  =/=  (/)
197, 8, 18w3a 936 . . . 4  wff  ( x  =/=  (/)  /\  (/)  e/  x  /\  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P (
y  i^i  z )
)  =/=  (/) )
202cv 1623 . . . . . 6  class  w
2120cpw 3627 . . . . 5  class  ~P w
2221cpw 3627 . . . 4  class  ~P ~P w
2319, 4, 22crab 2549 . . 3  class  { x  e.  ~P ~P w  |  ( x  =/=  (/)  /\  (/)  e/  x  /\  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P (
y  i^i  z )
)  =/=  (/) ) }
242, 3, 23cmpt 4079 . 2  class  ( w  e.  _V  |->  { x  e.  ~P ~P w  |  ( x  =/=  (/)  /\  (/)  e/  x  /\  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P (
y  i^i  z )
)  =/=  (/) ) } )
251, 24wceq 1624 1  wff  fBas  =  ( w  e.  _V  |->  { x  e.  ~P ~P w  |  (
x  =/=  (/)  /\  (/)  e/  x  /\  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P (
y  i^i  z )
)  =/=  (/) ) } )
Colors of variables: wff set class
This definition is referenced by:  isfbas  17519
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