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Mirrors > Home > MPE Home > Th. List > df-fil | Structured version Visualization 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, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
df-fil |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfil 21649 | . 2 | |
2 | vz | . . 3 | |
3 | cvv 3200 | . . 3 | |
4 | vf | . . . . . . . . 9 | |
5 | 4 | cv 1482 | . . . . . . . 8 |
6 | vx | . . . . . . . . . 10 | |
7 | 6 | cv 1482 | . . . . . . . . 9 |
8 | 7 | cpw 4158 | . . . . . . . 8 |
9 | 5, 8 | cin 3573 | . . . . . . 7 |
10 | c0 3915 | . . . . . . 7 | |
11 | 9, 10 | wne 2794 | . . . . . 6 |
12 | 6, 4 | wel 1991 | . . . . . 6 |
13 | 11, 12 | wi 4 | . . . . 5 |
14 | 2 | cv 1482 | . . . . . 6 |
15 | 14 | cpw 4158 | . . . . 5 |
16 | 13, 6, 15 | wral 2912 | . . . 4 |
17 | cfbas 19734 | . . . . 5 | |
18 | 14, 17 | cfv 5888 | . . . 4 |
19 | 16, 4, 18 | crab 2916 | . . 3 |
20 | 2, 3, 19 | cmpt 4729 | . 2 |
21 | 1, 20 | wceq 1483 | 1 |
Colors of variables: wff setvar class |
This definition is referenced by: isfil 21651 filunirn 21686 |
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