Theorem cosn

Description: Composition with an ordered pair singleton. (Contributed by Zhi Wang, 6-Oct-2025)

	Ref	Expression
Assertion	cosn	|- ( ( B e. U /\ C e. V ) -> ( A o. { <. B , C >. } ) = ( { B } X. ( A " { C } ) ) )

Proof

Step	Hyp	Ref	Expression
1		xpsng	|- ( ( B e. U /\ C e. V ) -> ( { B } X. { C } ) = { <. B , C >. } )
2	1	coeq2d	|- ( ( B e. U /\ C e. V ) -> ( A o. ( { B } X. { C } ) ) = ( A o. { <. B , C >. } ) )
3		coxp	|- ( A o. ( { B } X. { C } ) ) = ( { B } X. ( A " { C } ) )
4	2 3	eqtr3di	|- ( ( B e. U /\ C e. V ) -> ( A o. { <. B , C >. } ) = ( { B } X. ( A " { C } ) ) )