Metamath Proof Explorer

Theorem tz6.12c

Description: Corollary of Theorem 6.12(1) of TakeutiZaring p. 27. (Contributed by NM, 30-Apr-2004) (Proof shortened by SN, 23-Dec-2024)

Ref Expression
Assertion tz6.12c 
|- ( E! y A F y -> ( ( F ` A ) = y <-> A F y ) )

Proof

Step Hyp Ref Expression
1 df-fv 
 |-  ( F ` A ) = ( iota y A F y )
2 1 eqeq1i 
 |-  ( ( F ` A ) = y <-> ( iota y A F y ) = y )
3 iota1 
 |-  ( E! y A F y -> ( A F y <-> ( iota y A F y ) = y ) )
4 2 3 bitr4id 
 |-  ( E! y A F y -> ( ( F ` A ) = y <-> A F y ) )