The binary relation $R:=\subseteq$ with $\operatorname{Domain}(R)=\operatorname{Codomain}(R)=\mathscr{P}(A)$?

To be strict, the situation may be even more general. Not only can we have different sets in domain and codomain, but they need not be powersets of other sets.

If $X,Y$ are any sets of sets (e.g., the infinite subsets of $\Bbb N$, the finite finite subsets of $\Bbb C$ with prime power cardinality, the compact intervals in $\Bbb R$, ...), we can investigate the subset relation between eleemtns of $X$ and $Y$. (Admittedly, in most cases we'd consider the case where $X=Y$, though). That is, the relation $R=\{\,\langle a,b\rangle\in X\times Y \mid a\subseteq b\,\}$. Thus we restrict $\subseteq$ not to elements of some powersets, but rather to elements of the (otherwise arbitrary) sets $X$ and $Y$. The $R$ above might be carefully denoted as $\subseteq_{X,Y}$. The whole point, however, is that this distinction can in practically all contexts be considered practically irrelevant.