Definition df-fi 9312

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 9315). (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 9311	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3438	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1540	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1540	. . . . . . 7 class 𝑦
8	7	cint 4900	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1541	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1540	. . . . . . 7 class 𝑥
11	10	cpw 4552	. . . . . 6 class 𝒫 𝑥
12		cfn 8881	. . . . . 6 class Fin
13	11, 12	cin 3898	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3058	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2712	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5177	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1541	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9313