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	|- prod_ k e. A B = prod_ k e. A ( _I ` B )

Proof

Step	Hyp	Ref	Expression
1		prodeq2ii	|- ( A. k e. A ( _I ` B ) = ( _I ` ( _I ` B ) ) -> prod_ k e. A B = prod_ k e. A ( _I ` B ) )
2		fvex	|- ( _I ` B ) e. _V
3		fvi	|- ( ( _I ` B ) e. _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 e. A -> ( _I ` B ) = ( _I ` ( _I ` B ) ) )
7	1 6	mprg	|- prod_ k e. A B = prod_ k e. A ( _I ` B )