I am wondering how to prove that the class $\{ a\,\times\, b\,|\,a\in A ,\, b\in B\}$, where $A$ and $B$ are fixed given sets, is a set.
Obviously, one could apply a certain subset axiom to a concrete set that contains every element describen in the class. However, I am having a struggle choosing the correct set.
In fact, I think that $$\mathcal{P}\bigg(\bigcup_{a\in A}\bigcup_{b\in B}\big(\mathcal{P}(\mathcal{P}(a\times b))\big)\bigg)$$
Should be good enough, for if $A$ and $B$ are sets, then $A\times B$ is a subset of $\mathcal{P}(\mathcal{P}(A\cup B))$, and the set given abouve should contain every element of every possible product of subsets of $A$ and $B$, or at least that is what I am thinking right now.
By the axiom schema of replacement, when $f$ is a function defined of a set $A$, $$\{f(a)\ |\ a\in A\}$$ is a set. So in your case you can take $f(a,b)=a\times b$ for $(a,b)\in A\times B$.