# What notation to use for case distinction in a set

I'm currently in the following framework: $$n \in \mathbb{N}$$, $$I \subseteq \{1, \dots, n\}$$ and $$X \subseteq \mathbb{R}$$. Now, I need to define a set which contains all vectors $$v \in \mathbb{R}^n$$ such that the components of $$v$$ with indices in $$I$$ are elements of $$X$$ and the other components are zero. My naive approach to define this set looks like this

$$\{(x_1, \dots, x_n) \in \mathbb{R}^n: x_i \in X, i \in I, x_j = 0, j \in \{1, \dots, n\} \setminus I\}.$$

I don't like this definition as it contains two indices $$i, j$$ for the components. Also it is not immediately clear that the cases are well defined (each index is covered by a case and no index appears in both terms). In a normal cases environment one could write "else" for the second case. However a cases environment does not look nice within the braces of the surrounding set.

Is there some notation I am missing? How can I write this set more clearly? Of course I could use a description of the set as I did in my introduction of this question.

You can describe it very simply if you do a little preliminary setting up. Let $$Y_i=\begin{cases}X,&\text{if }i\in I\\\{0\},&\text{otherwise;}\end{cases}$$ the desired set is then $$\prod\limits_{i=1}^nY_i$$.
$$\{\langle x_1,\ldots,x_n\rangle\in\Bbb R^n:x_i\in X\text{ for all }i\in I,\text{ and }x_i=0\text{ otherwise}\}\,.$$