Theorem cocnv1ss

Description: Characterizing when the composition of a converse with a class is included in another. (Contributed by BJ, 22-Oct-2026)

	Ref	Expression
Assertion	cocnv1ss	|- ( ( `' B o. A ) C_ C <-> A. x A. y A. z ( ( x A y /\ z B y ) -> x C z ) )

Proof

Step	Hyp	Ref	Expression
1		cotrg	|- ( ( `' B o. A ) C_ C <-> A. x A. y A. z ( ( x A y /\ y `' B z ) -> x C z ) )
2		vex	|- y e. _V
3		vex	|- z e. _V
4	2 3	brcnv	|- ( y `' B z <-> z B y )
5	4	anbi2i	|- ( ( x A y /\ y `' B z ) <-> ( x A y /\ z B y ) )
6	5	imbi1i	|- ( ( ( x A y /\ y `' B z ) -> x C z ) <-> ( ( x A y /\ z B y ) -> x C z ) )
7	6	3albii	|- ( A. x A. y A. z ( ( x A y /\ y `' B z ) -> x C z ) <-> A. x A. y A. z ( ( x A y /\ z B y ) -> x C z ) )
8	1 7	bitri	|- ( ( `' B o. A ) C_ C <-> A. x A. y A. z ( ( x A y /\ z B y ) -> x C z ) )