If every element of $A$ is equivalent (by an equivalence relation $R$) to some element of $B$
$\forall x\in A\ldotp \exists y \in B\ldotp x\,R\,y$
and vice versa
$\forall y\in B\ldotp \exists x \in A\ldotp x\,R\,y$
then what can I deduce about the relationship between $A$ and $B$? I certainly can't say that they're equal, but they would seem to be equivalent in some sense. Perhaps one could say something to do with bijections here? Or perhaps equivalence classes?
My motivation for asking is that I have a theorem of the form outlined above, and I would like to find a simpler way to state it. Originally my $R$ was just straightforward equality ($=$), and then I could just deduce $A = B$. Now my $R$ is no longer $=$, but I'd still like to keep the statement of my theorem nice and succinct if I can.