Definition df-fi 9427

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 9430). (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 9426	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3454	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1569	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1569	. . . . . . 7 class 𝑦
8	7	cint 4910	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1570	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1569	. . . . . . 7 class 𝑥
11	10	cpw 4560	. . . . . 6 class 𝒫 𝑥
12		cfn 8998	. . . . . 6 class Fin
13	11, 12	cin 4092	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3090	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2741	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5189	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1570	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is used by:  fival  9428