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Mirrors > Home > MPE Home > Th. List > dffil  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 17830  . 2  
2  vz  . . 3  
3  cvv 2916  . . 3  
4  vf  . . . . . . . . 9  
5  4  cv 1648  . . . . . . . 8 
6  vx  . . . . . . . . . 10  
7  6  cv 1648  . . . . . . . . 9 
8  7  cpw 3759  . . . . . . . 8 
9  5, 8  cin 3279  . . . . . . 7 
10  c0 3588  . . . . . . 7  
11  9, 10  wne 2567  . . . . . 6 
12  6, 4  wel 1722  . . . . . 6 
13  11, 12  wi 4  . . . . 5 
14  2  cv 1648  . . . . . 6 
15  14  cpw 3759  . . . . 5 
16  13, 6, 15  wral 2666  . . . 4 
17  cfbas 16644  . . . . 5  
18  14, 17  cfv 5413  . . . 4 
19  16, 4, 18  crab 2670  . . 3 
20  2, 3, 19  cmpt 4226  . 2 
21  1, 20  wceq 1649  1 
Colors of variables: wff set class 
This definition is referenced by: isfil 17832 filunirn 17867 
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