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 22453 | . 2 class Fil | |
2 | vz | . . 3 setvar 𝑧 | |
3 | cvv 3494 | . . 3 class V | |
4 | vf | . . . . . . . . 9 setvar 𝑓 | |
5 | 4 | cv 1536 | . . . . . . . 8 class 𝑓 |
6 | vx | . . . . . . . . . 10 setvar 𝑥 | |
7 | 6 | cv 1536 | . . . . . . . . 9 class 𝑥 |
8 | 7 | cpw 4539 | . . . . . . . 8 class 𝒫 𝑥 |
9 | 5, 8 | cin 3935 | . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥) |
10 | c0 4291 | . . . . . . 7 class ∅ | |
11 | 9, 10 | wne 3016 | . . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅ |
12 | 6, 4 | wel 2115 | . . . . . 6 wff 𝑥 ∈ 𝑓 |
13 | 11, 12 | wi 4 | . . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) |
14 | 2 | cv 1536 | . . . . . 6 class 𝑧 |
15 | 14 | cpw 4539 | . . . . 5 class 𝒫 𝑧 |
16 | 13, 6, 15 | wral 3138 | . . . 4 wff ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) |
17 | cfbas 20533 | . . . . 5 class fBas | |
18 | 14, 17 | cfv 6355 | . . . 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 5146 | . 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) |
21 | 1, 20 | wceq 1537 | 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 22455 filunirn 22490 |
Copyright terms: Public domain | W3C validator |