Definition df-fi 9100

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 9103). (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 9099	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3422	. . 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 4876	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1539	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1538	. . . . . . 7 class 𝑥
11	10	cpw 4530	. . . . . 6 class 𝒫 𝑥
12		cfn 8691	. . . . . 6 class Fin
13	11, 12	cin 3882	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3064	. . . 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 5153	. 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  9101