In general how does one formulate a proof by controposition or contradiction for the following general form: $\forall x\exists ! y (P(x)\wedge Q(y) \rightarrow R(x,y))$
Or more specifically: $\forall x\exists ! y (x\geq 0 \wedge y\geq 0 \rightarrow (y^2\leq x<(y+1)^2))$
EDIT: $x$ and $y$ are (non-negative) integers.
If it makes it easier to disregard the uniqueness, I would be interested in that as well. FYI: I know how to prove it directly by using the floor function to define $y$, I just was wondering how it would be done indirectly.