Definition df-fi 9451

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 9454). (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 9450	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3480	. . 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 4946	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1540	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1539	. . . . . . 7 class 𝑥
11	10	cpw 4600	. . . . . 6 class 𝒫 𝑥
12		cfn 8985	. . . . . 6 class Fin
13	11, 12	cin 3950	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3070	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2714	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5225	. 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  9452