Definition df-fi 9295

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 9298). (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 9294	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3436	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1540	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1540	. . . . . . 7 class 𝑦
8	7	cint 4895	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1541	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1540	. . . . . . 7 class 𝑥
11	10	cpw 4547	. . . . . 6 class 𝒫 𝑥
12		cfn 8869	. . . . . 6 class Fin
13	11, 12	cin 3896	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3056	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2709	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5170	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1541	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9296