Restriction of a multivariate function

Question

What mathematical notation is most appropriate for restricting (or describing) the domain of a multivariate function? Suppose I want to restrict the domain of $f(x,y)$ so that $x$ is a real number from $10$ to $20$ and $y$ is an integer number from $-1$ to $10$. What alternatives do I have?

Answer 1

For merely describing the working domain of a function $f(x,y)$, maybe:

$\text{dom}f=\{x,y\vert x\in \mathbb{R}\wedge[10,20],y\in \mathbb{Z}\wedge y\in[-1,10] \}$

or, perhaps, more precisely:

$\text{dom}f=\{x\vert x\in \mathbb{R}\wedge[10,20]\}*\{y\vert y\in \mathbb{Z}\wedge y\in[-1,10]\}$

For intentionally restricting the domain, maybe...

$f\vert _D$ where $D=\{x\vert x\in \mathbb{R}\wedge[10,20]\}*\{y\vert y\in \mathbb{Z}\wedge y\in[-1,10]\}$

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