Theorem brab2ddw2

Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by Zhi Wang, 24-Sep-2025)

	Ref	Expression
Hypotheses	brab2dd.1	|- ( ph -> R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ps ) } )
	brab2ddw.2	|- ( x = A -> ( ps <-> th ) )
	brab2ddw.3	|- ( y = B -> ( th <-> ch ) )
	brab2ddw2.4	|- ( x = A -> C = U )
	brab2ddw2.5	|- ( y = B -> D = V )
Assertion	brab2ddw2	|- ( ph -> ( A R B <-> ( ( A e. U /\ B e. V ) /\ ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		brab2dd.1	|- ( ph -> R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ps ) } )
2		brab2ddw.2	|- ( x = A -> ( ps <-> th ) )
3		brab2ddw.3	|- ( y = B -> ( th <-> ch ) )
4		brab2ddw2.4	|- ( x = A -> C = U )
5		brab2ddw2.5	|- ( y = B -> D = V )
6	2 3	sylan9bb	|- ( ( x = A /\ y = B ) -> ( ps <-> ch ) )
7	6	adantl	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
8		id	|- ( x = A -> x = A )
9	8 4	eleq12d	|- ( x = A -> ( x e. C <-> A e. U ) )
10		id	|- ( y = B -> y = B )
11	10 5	eleq12d	|- ( y = B -> ( y e. D <-> B e. V ) )
12	9 11	bi2anan9	|- ( ( x = A /\ y = B ) -> ( ( x e. C /\ y e. D ) <-> ( A e. U /\ B e. V ) ) )
13	12	adantl	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( x e. C /\ y e. D ) <-> ( A e. U /\ B e. V ) ) )
14	1 7 13	brab2dd	|- ( ph -> ( A R B <-> ( ( A e. U /\ B e. V ) /\ ch ) ) )