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Definition df-fil 23332
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 23331 . 2 class Fil
2 vz . . 3 setvar 𝑧
3 cvv 3475 . . 3 class V
4 vf . . . . . . . . 9 setvar 𝑓
54cv 1541 . . . . . . . 8 class 𝑓
6 vx . . . . . . . . . 10 setvar 𝑥
76cv 1541 . . . . . . . . 9 class 𝑥
87cpw 4601 . . . . . . . 8 class 𝒫 𝑥
95, 8cin 3946 . . . . . . 7 class (𝑓 ∩ 𝒫 𝑥)
10 c0 4321 . . . . . . 7 class
119, 10wne 2941 . . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅
126, 4wel 2108 . . . . . 6 wff 𝑥𝑓
1311, 12wi 4 . . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
142cv 1541 . . . . . 6 class 𝑧
1514cpw 4601 . . . . 5 class 𝒫 𝑧
1613, 6, 15wral 3062 . . . 4 wff 𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)
17 cfbas 20917 . . . . 5 class fBas
1814, 17cfv 6540 . . . 4 class (fBas‘𝑧)
1916, 4, 18crab 3433 . . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)}
202, 3, 19cmpt 5230 . 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
211, 20wceq 1542 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  23333  filunirn  23368
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