![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23868 | . 2 class Fil | |
2 | vz | . . 3 setvar 𝑧 | |
3 | cvv 3477 | . . 3 class V | |
4 | vf | . . . . . . . . 9 setvar 𝑓 | |
5 | 4 | cv 1535 | . . . . . . . 8 class 𝑓 |
6 | vx | . . . . . . . . . 10 setvar 𝑥 | |
7 | 6 | cv 1535 | . . . . . . . . 9 class 𝑥 |
8 | 7 | cpw 4604 | . . . . . . . 8 class 𝒫 𝑥 |
9 | 5, 8 | cin 3961 | . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥) |
10 | c0 4338 | . . . . . . 7 class ∅ | |
11 | 9, 10 | wne 2937 | . . . . . 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 1535 | . . . . . 6 class 𝑧 |
15 | 14 | cpw 4604 | . . . . 5 class 𝒫 𝑧 |
16 | 13, 6, 15 | wral 3058 | . . . 4 wff ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) |
17 | cfbas 21369 | . . . . 5 class fBas | |
18 | 14, 17 | cfv 6562 | . . . 4 class (fBas‘𝑧) |
19 | 16, 4, 18 | crab 3432 | . . 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 1536 | 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 23870 filunirn 23905 |
Copyright terms: Public domain | W3C validator |