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Mirrors > Home > MPE Home > Th. List > df-fil | Structured version Visualization version GIF 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 ℝ. 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 | ⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfil 23331 | . 2 class Fil | |
2 | vz | . . 3 setvar 𝑧 | |
3 | cvv 3475 | . . 3 class V | |
4 | vf | . . . . . . . . 9 setvar 𝑓 | |
5 | 4 | cv 1541 | . . . . . . . 8 class 𝑓 |
6 | vx | . . . . . . . . . 10 setvar 𝑥 | |
7 | 6 | cv 1541 | . . . . . . . . 9 class 𝑥 |
8 | 7 | cpw 4601 | . . . . . . . 8 class 𝒫 𝑥 |
9 | 5, 8 | cin 3946 | . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥) |
10 | c0 4321 | . . . . . . 7 class ∅ | |
11 | 9, 10 | wne 2941 | . . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅ |
12 | 6, 4 | wel 2108 | . . . . . 6 wff 𝑥 ∈ 𝑓 |
13 | 11, 12 | wi 4 | . . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) |
14 | 2 | cv 1541 | . . . . . 6 class 𝑧 |
15 | 14 | cpw 4601 | . . . . 5 class 𝒫 𝑧 |
16 | 13, 6, 15 | wral 3062 | . . . 4 wff ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) |
17 | cfbas 20917 | . . . . 5 class fBas | |
18 | 14, 17 | cfv 6540 | . . . 4 class (fBas‘𝑧) |
19 | 16, 4, 18 | crab 3433 | . . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)} |
20 | 2, 3, 19 | cmpt 5230 | . 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) |
21 | 1, 20 | wceq 1542 | 1 wff Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) |
Colors of variables: wff setvar class |
This definition is referenced by: isfil 23333 filunirn 23368 |
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