Theorem sumfc

Description: A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013) (Revised by Mario Carneiro, 23-Apr-2014)

	Ref	Expression
Assertion	sumfc	|- sum_ j e. A ( ( k e. A |-> B ) ` j ) = sum_ 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	sumeq2i	|- sum_ k e. A ( ( k e. A |-> B ) ` k ) = sum_ 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	cbvsumi	|- sum_ j e. A ( ( k e. A |-> B ) ` j ) = sum_ k e. A ( ( k e. A |-> B ) ` k )
8		sum2id	|- sum_ k e. A B = sum_ k e. A ( _I ` B )
9	3 7 8	3eqtr4i	|- sum_ j e. A ( ( k e. A |-> B ) ` j ) = sum_ k e. A B