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Metamath Proof Explorer

Theorem fnrnov

Description: The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006)

Ref Expression
Assertion fnrnov 
|- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )

Proof

Step Hyp Ref Expression
1 fnrnfv 
 |-  ( F Fn ( A X. B ) -> ran F = { z | E. w e. ( A X. B ) z = ( F ` w ) } )
2 fveq2 
 |-  ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3 df-ov 
 |-  ( x F y ) = ( F ` <. x , y >. )
4 2 3 eqtr4di 
 |-  ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5 4 eqeq2d 
 |-  ( w = <. x , y >. -> ( z = ( F ` w ) <-> z = ( x F y ) ) )
6 5 rexxp 
 |-  ( E. w e. ( A X. B ) z = ( F ` w ) <-> E. x e. A E. y e. B z = ( x F y ) )
7 6 abbii 
 |-  { z | E. w e. ( A X. B ) z = ( F ` w ) } = { z | E. x e. A E. y e. B z = ( x F y ) }
8 1 7 eqtrdi 
 |-  ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )