Definition df-fi 9324

Description: Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 9327). (Contributed by FL, 27-Apr-2008.)

Assertion

Ref	Expression
df-fi	⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-fi

Step	Hyp	Ref	Expression
1		cfi 9323	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3429	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1541	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1541	. . . . . . 7 class 𝑦
8	7	cint 4889	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1542	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1541	. . . . . . 7 class 𝑥
11	10	cpw 4541	. . . . . 6 class 𝒫 𝑥
12		cfn 8893	. . . . . 6 class Fin
13	11, 12	cin 3888	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3061	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2714	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5166	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1542	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9325