Definition df-fi 8859

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 8862). (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 8858	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3441	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1537	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1537	. . . . . . 7 class 𝑦
8	7	cint 4838	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1538	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1537	. . . . . . 7 class 𝑥
11	10	cpw 4497	. . . . . 6 class 𝒫 𝑥
12		cfn 8492	. . . . . 6 class Fin
13	11, 12	cin 3880	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3107	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2776	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5110	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1538	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  8860