Theorem cosnop

Description: Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023)

	Ref	Expression
Hypotheses	cosnop.a	|- ( ph -> A e. V )
	cosnop.b	|- ( ph -> B e. W )
	cosnop.c	|- ( ph -> C e. X )
Assertion	cosnop	|- ( ph -> ( { <. A , B >. } o. { <. C , A >. } ) = { <. C , B >. } )

Proof

Step	Hyp	Ref	Expression
1		cosnop.a	|- ( ph -> A e. V )
2		cosnop.b	|- ( ph -> B e. W )
3		cosnop.c	|- ( ph -> C e. X )
4		snnzg	|- ( A e. V -> { A } =/= (/) )
5		xpco	|- ( { A } =/= (/) -> ( ( { A } X. { B } ) o. ( { C } X. { A } ) ) = ( { C } X. { B } ) )
6	1 4 5	3syl	|- ( ph -> ( ( { A } X. { B } ) o. ( { C } X. { A } ) ) = ( { C } X. { B } ) )
7		xpsng	|- ( ( A e. V /\ B e. W ) -> ( { A } X. { B } ) = { <. A , B >. } )
8	1 2 7	syl2anc	|- ( ph -> ( { A } X. { B } ) = { <. A , B >. } )
9		xpsng	|- ( ( C e. X /\ A e. V ) -> ( { C } X. { A } ) = { <. C , A >. } )
10	3 1 9	syl2anc	|- ( ph -> ( { C } X. { A } ) = { <. C , A >. } )
11	8 10	coeq12d	|- ( ph -> ( ( { A } X. { B } ) o. ( { C } X. { A } ) ) = ( { <. A , B >. } o. { <. C , A >. } ) )
12		xpsng	|- ( ( C e. X /\ B e. W ) -> ( { C } X. { B } ) = { <. C , B >. } )
13	3 2 12	syl2anc	|- ( ph -> ( { C } X. { B } ) = { <. C , B >. } )
14	6 11 13	3eqtr3d	|- ( ph -> ( { <. A , B >. } o. { <. C , A >. } ) = { <. C , B >. } )