The following is a problem question in How to Prove It (4.1.14b):
For each $(i, j) ∈ I × I$ let $C(i,,j) = A_i × B_j$, and let $P = I × I$. Prove that $∪_{p∈P}C_p = (∪_{i∈I} A_i)\times (∪_{i∈I} B_i)$
My question is regarding indexing. Should $y\in ∪_{j∈I} B_j $ instead? Yes, i is a free variable, but it shouldn't be used twice here since it represents the index number for A which isn't necessarily the same as the one used for B. Am I correct? Can someone clarify
The use of $i$ as an index for $\bigcup_{i \in I} A_i$ ends, in essence, with the parentheticals around that symbol. After all, in a similar vein, $$ \bigcup_{i=1}^{100} A_i $$ is just essentially a shorthand for the expression $$ A_1 \cup A_2 \cup \cdots \cup A_{100} $$ If something else does not depend on the running index, you're allowed to use it again elsewhere, which can be nice for homogenity and consistency and to keep things in line with certain naming conventions.
To poke at this notion of dependence, note the difference between $$ \left( \bigcup_{i=1}^{100} A_i \right) \cup \left( \bigcup_{i=1}^{100} B_i \right) $$ and, say, $$ \bigcup_{i=1}^{100} \left( A_i \cup \bigcup_{k=i}^{100} B_k \right) $$ In the first, neither larger expression depends on $i$ in particular - again, it just serves the role of running index. In the second, on the other hand, the second union explicitly depends on the running index $i$ in question. This is a case where two separate running indices are needed.
I suppose a computer science analogy comes with "for" loops. The first of the two expressions above can be thought of two separate "for" loops for unions (which the result from each loop then being unioned together). Many programming languages would have no issue with you using the same indexing variable for each. On the other second, for the second expression, you would have a for loop nested within another for loop, so you cannot (usually) just naively use the "outer" indexing variable.
Now, that said, you don't have to do this. If you please, you may write $$ \left( \bigcup_{i \in I} A_i \right) \times \left( \bigcup_{j \in I} B_j \right) $$ if you want, with no issues. For a problem phrased as the one given (where clear roles for $i,j$ are given by how $C(\cdot,\cdot)$ is defined), I'd say it's even more natural.