Metamath Proof Explorer

Theorem tz6.12

Description: Function value. Theorem 6.12(1) of TakeutiZaring p. 27. (Contributed by NM, 10-Jul-1994)

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

Proof

Step Hyp Ref Expression
1 df-br 
 |-  ( A F y <-> <. A , y >. e. F )
2 1 eubii 
 |-  ( E! y A F y <-> E! y <. A , y >. e. F )
3 tz6.12-1 
 |-  ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )
4 1 2 3 syl2anbr 
 |-  ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )