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	$$⊢\left(⟨A,y⟩\in F\wedge \exists !y⟨A,y⟩\in F\right)\to F\left(A\right)=y$$

Proof

Step	Hyp	Ref	Expression
1		df-br	$$⊢AFy\leftrightarrow ⟨A,y⟩\in F$$
2	1	eubii	$$⊢\exists !yAFy\leftrightarrow \exists !y⟨A,y⟩\in F$$
3		tz6.12-1	$$⊢\left(AFy\wedge \exists !yAFy\right)\to F\left(A\right)=y$$
4	1 2 3	syl2anbr	$$⊢\left(⟨A,y⟩\in F\wedge \exists !y⟨A,y⟩\in F\right)\to F\left(A\right)=y$$