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Definition df-fil 21859
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 21858 . 2 class Fil
2 vz . . 3 setvar 𝑧
3 cvv 3391 . . 3 class V
4 vf . . . . . . . . 9 setvar 𝑓
54cv 1636 . . . . . . . 8 class 𝑓
6 vx . . . . . . . . . 10 setvar 𝑥
76cv 1636 . . . . . . . . 9 class 𝑥
87cpw 4351 . . . . . . . 8 class 𝒫 𝑥
95, 8cin 3768 . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥)
10 c0 4116 . . . . . . 7 class
119, 10wne 2978 . . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅
126, 4wel 2157 . . . . . 6 wff 𝑥𝑓
1311, 12wi 4 . . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
142cv 1636 . . . . . 6 class 𝑧
1514cpw 4351 . . . . 5 class 𝒫 𝑧
1613, 6, 15wral 3096 . . . 4 wff 𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
17 cfbas 19938 . . . . 5 class fBas
1814, 17cfv 6097 . . . 4 class (fBas‘𝑧)
1916, 4, 18crab 3100 . . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)}
202, 3, 19cmpt 4923 . 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
211, 20wceq 1637 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  21860  filunirn  21895
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