Metamath Proof Explorer

Theorem tz6.12-1

Description: Function value. Theorem 6.12(1) of TakeutiZaring p. 27. (Contributed by NM, 30-Apr-2004)

Ref Expression
Assertion tz6.12-1 
|- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )

Proof

Step Hyp Ref Expression
1 df-fv 
 |-  ( F ` A ) = ( iota y A F y )
2 iota1 
 |-  ( E! y A F y -> ( A F y <-> ( iota y A F y ) = y ) )
3 2 biimpac 
 |-  ( ( A F y /\ E! y A F y ) -> ( iota y A F y ) = y )
4 1 3 eqtrid 
 |-  ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )