Metamath Proof Explorer


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 jAkABj=kAB

Proof

Step Hyp Ref Expression
1 eqid kAB=kAB
2 1 fvmpt2i kAkABk=IB
3 2 sumeq2i kAkABk=kAIB
4 nffvmpt1 _kkABj
5 nfcv _jkABk
6 fveq2 j=kkABj=kABk
7 4 5 6 cbvsumi jAkABj=kAkABk
8 sum2id kAB=kAIB
9 3 7 8 3eqtr4i jAkABj=kAB