Definition df-fi 9168

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 9171). (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 9167	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3431	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1538	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1538	. . . . . . 7 class 𝑦
8	7	cint 4881	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1539	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1538	. . . . . . 7 class 𝑥
11	10	cpw 4535	. . . . . 6 class 𝒫 𝑥
12		cfn 8731	. . . . . 6 class Fin
13	11, 12	cin 3887	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3065	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2715	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5159	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1539	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9169