Let $\mathcal{I}$ be an index set, and $E$ another set.
I have a set $A$ of tuples which I don't know how to express. Each contains both a subset of $\mathcal{I}$ ("domains") and a family of values in $E$ indexed by this very subset:
\begin{align} \forall a = (a_0,\ a_1) \in A,\quad a_0 &\subset \mathcal{I}\\ a_1 &= (v_i \in E)_{i \in a_0} \end{align}
How to write $A$ then?
$A = 2^\mathcal{I} \times ?$
$A = 2^\mathcal{I} \times E^\mathcal{I}\ ?$
$A = 2^\mathcal{I} \times E_\mathcal{I}\ ?$
$A = \mathcal{I} : E_{|\mathcal{I}}\ ?$