Metamath Proof Explorer


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 )