Theorem sum2id

Description: The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005) (Revised by Mario Carneiro, 13-Jul-2013)

	Ref	Expression
Assertion	sum2id	⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sumeq2ii	⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ ( I ‘ 𝐵 ) ) → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) )
2		fvex	⊢ ( I ‘ 𝐵 ) ∈ V
3		fvi	⊢ ( ( I ‘ 𝐵 ) ∈ V → ( I ‘ ( I ‘ 𝐵 ) ) = ( I ‘ 𝐵 ) )
4	2 3	ax-mp	⊢ ( I ‘ ( I ‘ 𝐵 ) ) = ( I ‘ 𝐵 )
5	4	eqcomi	⊢ ( I ‘ 𝐵 ) = ( I ‘ ( I ‘ 𝐵 ) )
6	5	a1i	⊢ ( 𝑘 ∈ 𝐴 → ( I ‘ 𝐵 ) = ( I ‘ ( I ‘ 𝐵 ) ) )
7	1 6	mprg	⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 )