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

Proof

Step	Hyp	Ref	Expression
1		df-fv	$$⊢F\left(A\right)=\left(\iota y|AFy\right)$$
2	1	eqeq1i	$$⊢F\left(A\right)=y\leftrightarrow \left(\iota y|AFy\right)=y$$
3		iota1	$$⊢\exists !yAFy\to \left(AFy\leftrightarrow \left(\iota y|AFy\right)=y\right)$$
4	2 3	bitr4id	$$⊢\exists !yAFy\to \left(F\left(A\right)=y\leftrightarrow AFy\right)$$