Definition df-fi 9359

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 9362). (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 9358	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3456	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1561	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1561	. . . . . . 7 class 𝑦
8	7	cint 4907	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1562	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1561	. . . . . . 7 class 𝑥
11	10	cpw 4557	. . . . . 6 class 𝒫 𝑥
12		cfn 8929	. . . . . 6 class Fin
13	11, 12	cin 3905	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3088	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2742	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5183	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1562	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9360