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	$$⊢\left(AFy\wedge \exists !yAFy\right)\to F\left(A\right)=y$$

Proof

Step	Hyp	Ref	Expression
1		df-fv	$$⊢F\left(A\right)=\left(\iota y|AFy\right)$$
2		iota1	$$⊢\exists !yAFy\to \left(AFy\leftrightarrow \left(\iota y|AFy\right)=y\right)$$
3	2	biimpac	$$⊢\left(AFy\wedge \exists !yAFy\right)\to \left(\iota y|AFy\right)=y$$
4	1 3	syl5eq	$$⊢\left(AFy\wedge \exists !yAFy\right)\to F\left(A\right)=y$$