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Metamath Proof Explorer

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