Definition df-fi 6967

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 6970). (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 6966	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2737	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1352	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1352	. . . . . . 7 class 𝑦
8	7	cint 3844	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1353	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1352	. . . . . . 7 class 𝑥
11	10	cpw 3575	. . . . . 6 class 𝒫 𝑥
12		cfn 6739	. . . . . 6 class Fin
13	11, 12	cin 3128	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2456	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2163	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4064	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1353	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  6968