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

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( k e. A |-> B ) = ( k e. A |-> B )
2	1	fvmpt2i	|- ( k e. A -> ( ( k e. A |-> B ) ` k ) = ( _I ` B ) )
3	2	prodeq2i	|- prod_ k e. A ( ( k e. A |-> B ) ` k ) = prod_ k e. A ( _I ` B )
4		nffvmpt1	|- F/_ k ( ( k e. A |-> B ) ` j )
5		nfcv	|- F/_ j ( ( k e. A |-> B ) ` k )
6		fveq2	|- ( j = k -> ( ( k e. A |-> B ) ` j ) = ( ( k e. A |-> B ) ` k ) )
7	4 5 6	cbvprodi	|- prod_ j e. A ( ( k e. A |-> B ) ` j ) = prod_ k e. A ( ( k e. A |-> B ) ` k )
8		prod2id	|- prod_ k e. A B = prod_ k e. A ( _I ` B )
9	3 7 8	3eqtr4i	|- prod_ j e. A ( ( k e. A |-> B ) ` j ) = prod_ k e. A B