Definition df-fi 9421

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 9424). (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 9420	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3459	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1539	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1539	. . . . . . 7 class 𝑦
8	7	cint 4922	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1540	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1539	. . . . . . 7 class 𝑥
11	10	cpw 4575	. . . . . 6 class 𝒫 𝑥
12		cfn 8957	. . . . . 6 class Fin
13	11, 12	cin 3925	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3060	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2713	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5201	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1540	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9422