Metamath Proof Explorer

Theorem rnxrnres

Description: Range of a range Cartesian product with a restricted relation. (Contributed by Peter Mazsa, 5-Dec-2021)

Ref Expression
Assertion rnxrnres 
|- ran ( R |X. ( S |` A ) ) = { <. x , y >. | E. u e. A ( u R x /\ u S y ) }

Proof

Step Hyp Ref Expression
1 rnxrn 
 |-  ran ( R |X. ( S |` A ) ) = { <. x , y >. | E. u ( u R x /\ u ( S |` A ) y ) }
2 brres 
 |-  ( y e. _V -> ( u ( S |` A ) y <-> ( u e. A /\ u S y ) ) )
3 2 elv 
 |-  ( u ( S |` A ) y <-> ( u e. A /\ u S y ) )
4 3 anbi2i 
 |-  ( ( u R x /\ u ( S |` A ) y ) <-> ( u R x /\ ( u e. A /\ u S y ) ) )
5 an12 
 |-  ( ( u e. A /\ ( u R x /\ u S y ) ) <-> ( u R x /\ ( u e. A /\ u S y ) ) )
6 4 5 bitr4i 
 |-  ( ( u R x /\ u ( S |` A ) y ) <-> ( u e. A /\ ( u R x /\ u S y ) ) )
7 6 exbii 
 |-  ( E. u ( u R x /\ u ( S |` A ) y ) <-> E. u ( u e. A /\ ( u R x /\ u S y ) ) )
8 df-rex 
 |-  ( E. u e. A ( u R x /\ u S y ) <-> E. u ( u e. A /\ ( u R x /\ u S y ) ) )
9 7 8 bitr4i 
 |-  ( E. u ( u R x /\ u ( S |` A ) y ) <-> E. u e. A ( u R x /\ u S y ) )
10 9 opabbii 
 |-  { <. x , y >. | E. u ( u R x /\ u ( S |` A ) y ) } = { <. x , y >. | E. u e. A ( u R x /\ u S y ) }
11 1 10 eqtri 
 |-  ran ( R |X. ( S |` A ) ) = { <. x , y >. | E. u e. A ( u R x /\ u S y ) }