Theorem cocnv1ssid

Description: Characterizing when the composition of a converse with a class is included in the identity relation. The instance where the two classes are equal gives a characterization of injectivity (see df-f1 ). (Contributed by BJ, 22-Oct-2026)

	Ref	Expression
Assertion	cocnv1ssid	|- ( ( `' B o. A ) C_ _I <-> A. x A. y A. z ( ( x A y /\ z B y ) -> x = z ) )

Proof

Step	Hyp	Ref	Expression
1		cocnv1ss	|- ( ( `' B o. A ) C_ _I <-> A. x A. y A. z ( ( x A y /\ z B y ) -> x _I z ) )
2		vex	|- z e. _V
3	2	ideq	|- ( x _I z <-> x = z )
4	3	imbi2i	|- ( ( ( x A y /\ z B y ) -> x _I z ) <-> ( ( x A y /\ z B y ) -> x = z ) )
5	4	3albii	|- ( A. x A. y A. z ( ( x A y /\ z B y ) -> x _I z ) <-> A. x A. y A. z ( ( x A y /\ z B y ) -> x = z ) )
6	1 5	bitri	|- ( ( `' B o. A ) C_ _I <-> A. x A. y A. z ( ( x A y /\ z B y ) -> x = z ) )