Definition df-fil 22697

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 22696	. 2 class Fil
2		vz	. . 3 setvar 𝑧
3		cvv 3398	. . 3 class V
4		vf	. . . . . . . . 9 setvar 𝑓
5	4	cv 1542	. . . . . . . 8 class 𝑓
6		vx	. . . . . . . . . 10 setvar 𝑥
7	6	cv 1542	. . . . . . . . 9 class 𝑥
8	7	cpw 4499	. . . . . . . 8 class 𝒫 𝑥
9	5, 8	cin 3852	. . . . . . 7 class (𝑓 ∩ 𝒫 𝑥)
10		c0 4223	. . . . . . 7 class ∅
11	9, 10	wne 2932	. . . . . 6 wff (𝑓 ∩ 𝒫 𝑥) ≠ ∅
12	6, 4	wel 2113	. . . . . 6 wff 𝑥 ∈ 𝑓
13	11, 12	wi 4	. . . . 5 wff ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)
14	2	cv 1542	. . . . . 6 class 𝑧
15	14	cpw 4499	. . . . 5 class 𝒫 𝑧
16	13, 6, 15	wral 3051	. . . 4 wff ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)
17		cfbas 20305	. . . . 5 class fBas
18	14, 17	cfv 6358	. . . 4 class (fBas‘𝑧)
19	16, 4, 18	crab 3055	. . 3 class {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}
20	2, 3, 19	cmpt 5120	. 2 class (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
21	1, 20	wceq 1543	1 wff Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})

This definition is referenced by:  isfil  22698  filunirn  22733