Is this definition of Cartesian Products of Families correct?

Question

I was wondering if the way I defined the Cartesian Product of Families correct. Now, I am more inclined to predicate logic notation so if there may be any notational discrepancies, you must excuse me.

∏_{i ∈ I}X_i = {f(i) : f ∈ (⋃_{i ∈ I}X_i)^I ∧ ∀f(i)[f(i) ∈ X_i]}

Answer 1

No. The quantifier $\forall f(i)$ does not make sense. You can only quantify over variables. Furthermore, the set contains the actual function $f$, not merely its value at a particular $i$. So the correct definition would be

$$\prod\limits_{i \in I} X_i = \{f \in (\bigcup\limits_{i \in I} X_i)^I \mid \forall i \in I (f(i) \in X_i)\}$$