Image of Element under Cartesian Product of Subsets

Theorem
Let $S$ and $T$ be sets.

Let $A \subseteq S$ and $B \subseteq T$.

Let $\RR$ be the relation defined by the Cartesian product $A \times B$.

Then:
 * $\forall x \in A: \map \RR x = B$

Proof
We have: