Definition df-fi 7035

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 7038). (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 7034	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2763	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1363	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1363	. . . . . . 7 class 𝑦
8	7	cint 3874	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1364	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1363	. . . . . . 7 class 𝑥
11	10	cpw 3605	. . . . . 6 class 𝒫 𝑥
12		cfn 6799	. . . . . 6 class Fin
13	11, 12	cin 3156	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2476	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2182	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4094	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1364	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  7036