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Definition df-fil 22451
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 22450 . 2 class Fil
2 vz . . 3 setvar 𝑧
3 cvv 3441 . . 3 class V
4 vf . . . . . . . . 9 setvar 𝑓
54cv 1537 . . . . . . . 8 class 𝑓
6 vx . . . . . . . . . 10 setvar 𝑥
76cv 1537 . . . . . . . . 9 class 𝑥
87cpw 4497 . . . . . . . 8 class 𝒫 𝑥
95, 8cin 3880 . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥)
10 c0 4243 . . . . . . 7 class
119, 10wne 2987 . . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅
126, 4wel 2112 . . . . . 6 wff 𝑥𝑓
1311, 12wi 4 . . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
142cv 1537 . . . . . 6 class 𝑧
1514cpw 4497 . . . . 5 class 𝒫 𝑧
1613, 6, 15wral 3106 . . . 4 wff 𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
17 cfbas 20079 . . . . 5 class fBas
1814, 17cfv 6324 . . . 4 class (fBas‘𝑧)
1916, 4, 18crab 3110 . . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)}
202, 3, 19cmpt 5110 . 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
211, 20wceq 1538 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  22452  filunirn  22487
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