Definition df-fi 6934

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 6937). (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 6933	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2726	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1342	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1342	. . . . . . 7 class 𝑦
8	7	cint 3824	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1343	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1342	. . . . . . 7 class 𝑥
11	10	cpw 3559	. . . . . 6 class 𝒫 𝑥
12		cfn 6706	. . . . . 6 class Fin
13	11, 12	cin 3115	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2445	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2151	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4043	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1343	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  6935