Theorem opelopabd

Description: Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023)

	Ref	Expression
Hypotheses	opelopabd.xph	|- ( ph -> A. x ph )
	opelopabd.yph	|- ( ph -> A. y ph )
	opelopabd.xch	|- ( ph -> F/ x ch )
	opelopabd.ych	|- ( ph -> F/ y ch )
	opelopabd.exa	|- ( ph -> A e. U )
	opelopabd.exb	|- ( ph -> B e. V )
	opelopabd.is	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
Assertion	opelopabd	|- ( ph -> ( <. A , B >. e. { <. x , y >. | ps } <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		opelopabd.xph	|- ( ph -> A. x ph )
2		opelopabd.yph	|- ( ph -> A. y ph )
3		opelopabd.xch	|- ( ph -> F/ x ch )
4		opelopabd.ych	|- ( ph -> F/ y ch )
5		opelopabd.exa	|- ( ph -> A e. U )
6		opelopabd.exb	|- ( ph -> B e. V )
7		opelopabd.is	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
8		elopab	|- ( <. A , B >. e. { <. x , y >. | ps } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) )
9	1 2 3 4 5 6 7	copsex2d	|- ( ph -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) )
10	8 9	syl5bb	|- ( ph -> ( <. A , B >. e. { <. x , y >. | ps } <-> ch ) )