Definition df-fi 7303

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 7306). (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 7302	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2821	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1401	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1401	. . . . . . 7 class 𝑦
8	7	cint 3970	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1402	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1401	. . . . . . 7 class 𝑥
11	10	cpw 3688	. . . . . 6 class 𝒫 𝑥
12		cfn 7022	. . . . . 6 class Fin
13	11, 12	cin 3219	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2529	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2224	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4192	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1402	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is used by:  fival  7304