Definition df-fil 24165

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 ℝ. 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) ≠ ∅ → 𝑥 ∈ 𝑓)})

Distinct variable group:   𝑥,𝑓,𝑧

Detailed syntax breakdown of Definition df-fil

Step	Hyp	Ref	Expression
1		cfil 24164	. 2 class Fil
2		vz	. . 3 setvar 𝑧
3		cvv 3451	. . 3 class V
4		vf	. . . . . . . . 9 setvar 𝑓
5	4	cv 1569	. . . . . . . 8 class 𝑓
6		vx	. . . . . . . . . 10 setvar 𝑥
7	6	cv 1569	. . . . . . . . 9 class 𝑥
8	7	cpw 4557	. . . . . . . 8 class 𝒫 𝑥
9	5, 8	cin 3898	. . . . . . 7 class (𝑓 ∩ 𝒫 𝑥)
10		c0 4279	. . . . . . 7 class ∅
11	9, 10	wne 2956	. . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅
12	6, 4	wel 2146	. . . . . 6 wff 𝑥 ∈ 𝑓
13	11, 12	wi 4	. . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)
14	2	cv 1569	. . . . . 6 class 𝑧
15	14	cpw 4557	. . . . 5 class 𝒫 𝑧
16	13, 6, 15	wral 3077	. . . 4 wff ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)
17		cfbas 21666	. . . . 5 class fBas
18	14, 17	cfv 6538	. . . 4 class (fBas‘𝑧)
19	16, 4, 18	crab 3413	. . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}
20	2, 3, 19	cmpt 5186	. 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
21	1, 20	wceq 1570	1 wff Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})

This definition is used by:  isfil  24166  filunirn  24201