Definition df-fi 9179

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 9182). (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 9178	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3433	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1538	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1538	. . . . . . 7 class 𝑦
8	7	cint 4880	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1539	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1538	. . . . . . 7 class 𝑥
11	10	cpw 4534	. . . . . 6 class 𝒫 𝑥
12		cfn 8742	. . . . . 6 class Fin
13	11, 12	cin 3887	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3066	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2716	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5158	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1539	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9180