Definition df-fi 9480

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 9483). (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 9479	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3488	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1536	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1536	. . . . . . 7 class 𝑦
8	7	cint 4970	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1537	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1536	. . . . . . 7 class 𝑥
11	10	cpw 4622	. . . . . 6 class 𝒫 𝑥
12		cfn 9003	. . . . . 6 class Fin
13	11, 12	cin 3975	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3076	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2717	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5249	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1537	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9481