Definition df-fi 9373

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 9376). (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 9372	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3463	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1566	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1566	. . . . . . 7 class 𝑦
8	7	cint 4916	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1567	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1566	. . . . . . 7 class 𝑥
11	10	cpw 4567	. . . . . 6 class 𝒫 𝑥
12		cfn 8945	. . . . . 6 class Fin
13	11, 12	cin 3912	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3095	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2747	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5196	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1567	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9374