Theorem prodfc

Description: A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017)

	Ref	Expression
Assertion	prodfc	⊢ ∏ 𝑗 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = ∏ 𝑘 ∈ 𝐴 𝐵

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 )
2	1	fvmpt2i	⊢ ( 𝑘 ∈ 𝐴 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = ( I ‘ 𝐵 ) )
3	2	prodeq2i	⊢ ∏ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = ∏ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 )
4		nffvmpt1	⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 )
5		nfcv	⊢ Ⅎ 𝑗 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 )
6		fveq2	⊢ ( 𝑗 = 𝑘 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) )
7	4 5 6	cbvprodi	⊢ ∏ 𝑗 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = ∏ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 )
8		prod2id	⊢ ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 )
9	3 7 8	3eqtr4i	⊢ ∏ 𝑗 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = ∏ 𝑘 ∈ 𝐴 𝐵