Definition df-fi 9319

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 9322). (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 9318	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3430	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1541	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1541	. . . . . . 7 class 𝑦
8	7	cint 4890	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1542	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1541	. . . . . . 7 class 𝑥
11	10	cpw 4542	. . . . . 6 class 𝒫 𝑥
12		cfn 8888	. . . . . 6 class Fin
13	11, 12	cin 3889	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3062	. . . 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 5167	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1542	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9320