Definition df-fi 9362

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 9365). (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 9361	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3447	. . 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 4910	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1540	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1539	. . . . . . 7 class 𝑥
11	10	cpw 4563	. . . . . 6 class 𝒫 𝑥
12		cfn 8918	. . . . . 6 class Fin
13	11, 12	cin 3913	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3053	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2707	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5188	. 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  9363