df-fi

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 ). (Contributed by FL, 27-Apr-2008)

		Ref	Expression
	Assertion	df-fi	⊢ fi = ( 𝑥 ∈ V ↦ { 𝑧 ∣ ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑧 = ∩ 𝑦 } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cfi	⊢ fi
1		vx	⊢ 𝑥
2		cvv	⊢ V
3		vz	⊢ 𝑧
4		vy	⊢ 𝑦
5	1	cv	⊢ 𝑥
6	5	cpw	⊢ 𝒫 𝑥
7		cfn	⊢ Fin
8	6 7	cin	⊢ ( 𝒫 𝑥 ∩ Fin )
9	3	cv	⊢ 𝑧
10	4	cv	⊢ 𝑦
11	10	cint	⊢ ∩ 𝑦
12	9 11	wceq	⊢ 𝑧 = ∩ 𝑦
13	12 4 8	wrex	⊢ ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑧 = ∩ 𝑦
14	13 3	cab	⊢ { 𝑧 ∣ ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑧 = ∩ 𝑦 }
15	1 2 14	cmpt	⊢ ( 𝑥 ∈ V ↦ { 𝑧 ∣ ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑧 = ∩ 𝑦 } )
16	0 15	wceq	⊢ fi = ( 𝑥 ∈ V ↦ { 𝑧 ∣ ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑧 = ∩ 𝑦 } )

Metamath Proof Explorer

Definition df-fi

Detailed syntax breakdown