Theorem elco

Description: Membership in a composition. (Contributed by BJ, 16-Aug-2026)

	Ref	Expression
Assertion	elco	|- ( A e. ( C o. B ) <-> E. x E. y ( A = <. x , y >. /\ E. z ( x B z /\ z C y ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-co	|- ( C o. B ) = { <. x , y >. | E. z ( x B z /\ z C y ) }
2	1	eleq2i	|- ( A e. ( C o. B ) <-> A e. { <. x , y >. | E. z ( x B z /\ z C y ) } )
3		elopab	|- ( A e. { <. x , y >. | E. z ( x B z /\ z C y ) } <-> E. x E. y ( A = <. x , y >. /\ E. z ( x B z /\ z C y ) ) )
4	2 3	bitri	|- ( A e. ( C o. B ) <-> E. x E. y ( A = <. x , y >. /\ E. z ( x B z /\ z C y ) ) )