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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 22383 | . 2 class Fil | |
2 | vz | . . 3 setvar 𝑧 | |
3 | cvv 3495 | . . 3 class V | |
4 | vf | . . . . . . . . 9 setvar 𝑓 | |
5 | 4 | cv 1527 | . . . . . . . 8 class 𝑓 |
6 | vx | . . . . . . . . . 10 setvar 𝑥 | |
7 | 6 | cv 1527 | . . . . . . . . 9 class 𝑥 |
8 | 7 | cpw 4537 | . . . . . . . 8 class 𝒫 𝑥 |
9 | 5, 8 | cin 3934 | . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥) |
10 | c0 4290 | . . . . . . 7 class ∅ | |
11 | 9, 10 | wne 3016 | . . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅ |
12 | 6, 4 | wel 2106 | . . . . . 6 wff 𝑥 ∈ 𝑓 |
13 | 11, 12 | wi 4 | . . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) |
14 | 2 | cv 1527 | . . . . . 6 class 𝑧 |
15 | 14 | cpw 4537 | . . . . 5 class 𝒫 𝑧 |
16 | 13, 6, 15 | wral 3138 | . . . 4 wff ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) |
17 | cfbas 20463 | . . . . 5 class fBas | |
18 | 14, 17 | cfv 6349 | . . . 4 class (fBas‘𝑧) |
19 | 16, 4, 18 | crab 3142 | . . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)} |
20 | 2, 3, 19 | cmpt 5138 | . 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) |
21 | 1, 20 | wceq 1528 | 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 22385 filunirn 22420 |
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