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Metamath Proof Explorer

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 ) ) )