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Definition df-fil 21460
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 21459 . 2 class Fil
2 vz . . 3 setvar 𝑧
3 cvv 3173 . . 3 class V
4 vf . . . . . . . . 9 setvar 𝑓
54cv 1474 . . . . . . . 8 class 𝑓
6 vx . . . . . . . . . 10 setvar 𝑥
76cv 1474 . . . . . . . . 9 class 𝑥
87cpw 4108 . . . . . . . 8 class 𝒫 𝑥
95, 8cin 3539 . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥)
10 c0 3874 . . . . . . 7 class
119, 10wne 2780 . . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅
126, 4wel 1978 . . . . . 6 wff 𝑥𝑓
1311, 12wi 4 . . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
142cv 1474 . . . . . 6 class 𝑧
1514cpw 4108 . . . . 5 class 𝒫 𝑧
1613, 6, 15wral 2896 . . . 4 wff 𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
17 cfbas 19555 . . . . 5 class fBas
1814, 17cfv 5804 . . . 4 class (fBas‘𝑧)
1916, 4, 18crab 2900 . . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)}
202, 3, 19cmpt 4643 . 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
211, 20wceq 1475 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  21461  filunirn  21496
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