Theorem prod2id

Description: The second class argument to a product can be chosen so that it is always a set. (Contributed by Scott Fenton, 4-Dec-2017)

	Ref	Expression
Assertion	prod2id	⊢ ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		prodeq2ii	⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ ( I ‘ 𝐵 ) ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 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	⊢ ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 )