Theorem brabd

Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023)

	Ref	Expression
Hypotheses	brabd.exa	|- ( ph -> A e. U )
	brabd.exb	|- ( ph -> B e. V )
	brabd.def	|- ( ph -> R = { <. x , y >. | ps } )
	brabd.is	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
Assertion	brabd	|- ( ph -> ( A R B <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		brabd.exa	|- ( ph -> A e. U )
2		brabd.exb	|- ( ph -> B e. V )
3		brabd.def	|- ( ph -> R = { <. x , y >. | ps } )
4		brabd.is	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
5		ax-5	|- ( ph -> A. x ph )
6		ax-5	|- ( ph -> A. y ph )
7		nfvd	|- ( ph -> F/ x ch )
8		nfvd	|- ( ph -> F/ y ch )
9	5 6 7 8 1 2 3 4	brabd0	|- ( ph -> ( A R B <-> ch ) )