Theorem tz6.12-1-afv

Description: Function value (Theorem 6.12(1) of TakeutiZaring p. 27, analogous to tz6.12-1 . (Contributed by Alexander van der Vekens, 29-Nov-2017)

	Ref	Expression
Assertion	tz6.12-1-afv	$$⊢\left(AFy\wedge \exists !yAFy\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=y$$

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-afv	$$⊢\left(⟨A,y⟩\in F\wedge \exists !y⟨A,y⟩\in F\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=y$$
4	1 2 3	syl2anb	$$⊢\left(AFy\wedge \exists !yAFy\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=y$$