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Mirrors > Home > MPE Home > Th. List > dffil  Structured version Unicode version 
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 . 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, 20Jul2007.) (Revised by Stefan O'Rear, 28Jul2015.) 
Ref  Expression 

dffil 
Step  Hyp  Ref  Expression 

1  cfil 17882  . 2  
2  vz  . . 3  
3  cvv 2958  . . 3  
4  vf  . . . . . . . . 9  
5  4  cv 1652  . . . . . . . 8 
6  vx  . . . . . . . . . 10  
7  6  cv 1652  . . . . . . . . 9 
8  7  cpw 3801  . . . . . . . 8 
9  5, 8  cin 3321  . . . . . . 7 
10  c0 3630  . . . . . . 7  
11  9, 10  wne 2601  . . . . . 6 
12  6, 4  wel 1727  . . . . . 6 
13  11, 12  wi 4  . . . . 5 
14  2  cv 1652  . . . . . 6 
15  14  cpw 3801  . . . . 5 
16  13, 6, 15  wral 2707  . . . 4 
17  cfbas 16694  . . . . 5  
18  14, 17  cfv 5457  . . . 4 
19  16, 4, 18  crab 2711  . . 3 
20  2, 3, 19  cmpt 4269  . 2 
21  1, 20  wceq 1653  1 
Colors of variables: wff set class 
This definition is referenced by: isfil 17884 filunirn 17919 
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