Definition df-fi 9403

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 9406). (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 9402	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3450	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1568	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1568	. . . . . . 7 class 𝑦
8	7	cint 4906	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1569	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1568	. . . . . . 7 class 𝑥
11	10	cpw 4556	. . . . . 6 class 𝒫 𝑥
12		cfn 8973	. . . . . 6 class Fin
13	11, 12	cin 4089	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3086	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2738	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5185	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1569	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is used by:  fival  9404