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Definition df-fil 17637
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 17636 . 2  class  Fil
2 vz . . 3  set  z
3 cvv 2864 . . 3  class  _V
4 vf . . . . . . . . 9  set  f
54cv 1641 . . . . . . . 8  class  f
6 vx . . . . . . . . . 10  set  x
76cv 1641 . . . . . . . . 9  class  x
87cpw 3701 . . . . . . . 8  class  ~P x
95, 8cin 3227 . . . . . . 7  class  ( f  i^i  ~P x )
10 c0 3531 . . . . . . 7  class  (/)
119, 10wne 2521 . . . . . 6  wff  ( f  i^i  ~P x )  =/=  (/)
126, 4wel 1711 . . . . . 6  wff  x  e.  f
1311, 12wi 4 . . . . 5  wff  ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f )
142cv 1641 . . . . . 6  class  z
1514cpw 3701 . . . . 5  class  ~P z
1613, 6, 15wral 2619 . . . 4  wff  A. x  e.  ~P  z ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f )
17 cfbas 16465 . . . . 5  class  fBas
1814, 17cfv 5334 . . . 4  class  ( fBas `  z )
1916, 4, 18crab 2623 . . 3  class  { f  e.  ( fBas `  z
)  |  A. x  e.  ~P  z ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f ) }
202, 3, 19cmpt 4156 . 2  class  ( z  e.  _V  |->  { f  e.  ( fBas `  z
)  |  A. x  e.  ~P  z ( ( f  i^i  ~P x
)  =/=  (/)  ->  x  e.  f ) } )
211, 20wceq 1642 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  17638  filunirn  17673
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