Definition df-fi 9328

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 9331). (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 9327	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3442	. . 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 4904	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1542	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1541	. . . . . . 7 class 𝑥
11	10	cpw 4556	. . . . . 6 class 𝒫 𝑥
12		cfn 8897	. . . . . 6 class Fin
13	11, 12	cin 3902	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3062	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2715	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5181	. 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  9329