Definition df-fi 9448

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 9451). (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 9447	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3477	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1535	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1535	. . . . . . 7 class 𝑦
8	7	cint 4950	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1536	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1535	. . . . . . 7 class 𝑥
11	10	cpw 4604	. . . . . 6 class 𝒫 𝑥
12		cfn 8983	. . . . . 6 class Fin
13	11, 12	cin 3961	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3067	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2711	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5230	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1536	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9449