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Definition df-fil 17373
Description: The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in  RR. With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
df-fil  |-  Fil  =  ( z  e.  _V  |->  { f  e.  (
fBas `  z )  |  A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
Distinct variable group:    x, f, z

Detailed syntax breakdown of Definition df-fil
StepHypRef Expression
1 cfil 17372 . 2  class  Fil
2 vz . . 3  set  z
3 cvv 2727 . . 3  class  _V
4 vf . . . . . . . . 9  set  f
54cv 1618 . . . . . . . 8  class  f
6 vx . . . . . . . . . 10  set  x
76cv 1618 . . . . . . . . 9  class  x
87cpw 3530 . . . . . . . 8  class  ~P x
95, 8cin 3077 . . . . . . 7  class  ( f  i^i  ~P x )
10 c0 3362 . . . . . . 7  class  (/)
119, 10wne 2412 . . . . . 6  wff  ( f  i^i  ~P x )  =/=  (/)
126, 4wel 1622 . . . . . 6  wff  x  e.  f
1311, 12wi 6 . . . . 5  wff  ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f )
142cv 1618 . . . . . 6  class  z
1514cpw 3530 . . . . 5  class  ~P z
1613, 6, 15wral 2509 . . . 4  wff  A. x  e.  ~P  z ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f )
17 cfbas 17350 . . . . 5  class  fBas
1814, 17cfv 4592 . . . 4  class  ( fBas `  z )
1916, 4, 18crab 2512 . . 3  class  { f  e.  ( fBas `  z
)  |  A. x  e.  ~P  z ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f ) }
202, 3, 19cmpt 3974 . 2  class  ( z  e.  _V  |->  { f  e.  ( fBas `  z
)  |  A. x  e.  ~P  z ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f ) } )
211, 20wceq 1619 1  wff  Fil  =  ( z  e.  _V  |->  { f  e.  (
fBas `  z )  |  A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
Colors of variables: wff set class
This definition is referenced by:  isfil  17374  filunirn  17409
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