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Definition df-fil 21697
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.)
Assertion
Ref Expression
df-fil Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
Distinct variable group:   𝑥,𝑓,𝑧

Detailed syntax breakdown of Definition df-fil
StepHypRef Expression
1 cfil 21696 . 2 class Fil
2 vz . . 3 setvar 𝑧
3 cvv 3231 . . 3 class V
4 vf . . . . . . . . 9 setvar 𝑓
54cv 1522 . . . . . . . 8 class 𝑓
6 vx . . . . . . . . . 10 setvar 𝑥
76cv 1522 . . . . . . . . 9 class 𝑥
87cpw 4191 . . . . . . . 8 class 𝒫 𝑥
95, 8cin 3606 . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥)
10 c0 3948 . . . . . . 7 class
119, 10wne 2823 . . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅
126, 4wel 2031 . . . . . 6 wff 𝑥𝑓
1311, 12wi 4 . . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
142cv 1522 . . . . . 6 class 𝑧
1514cpw 4191 . . . . 5 class 𝒫 𝑧
1613, 6, 15wral 2941 . . . 4 wff 𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
17 cfbas 19782 . . . . 5 class fBas
1814, 17cfv 5926 . . . 4 class (fBas‘𝑧)
1916, 4, 18crab 2945 . . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)}
202, 3, 19cmpt 4762 . 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
211, 20wceq 1523 1 wff Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
Colors of variables: wff setvar class
This definition is referenced by:  isfil  21698  filunirn  21733
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