Definition df-fi 9318

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 9321). (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 9317	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3433	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1547	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1547	. . . . . . 7 class 𝑦
8	7	cint 4880	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1548	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1547	. . . . . . 7 class 𝑥
11	10	cpw 4532	. . . . . 6 class 𝒫 𝑥
12		cfn 8887	. . . . . 6 class Fin
13	11, 12	cin 3884	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3065	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2719	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5156	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1548	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9319