It is a standard exercise (see Jech's "Set Theory" Exercise 13.8) to prove that ordinal addition and multiplication are $\Delta_1$ expressible functions. The proof for addition comes from noting that $\alpha+\beta$ is order isomorphic to the disjoint union $(\{1\}\times \alpha)\cup (\{2\}\times \beta)$ under the lexicographical order (and everything after the "order isomorphic" part is all $\Delta_0$ expressible). Similarly $\alpha\cdot \beta$ is order isomorphic to the direct product $\beta\times \alpha$ under the lexicographical order.

It was surprised to see that ordinal exponentiation was not listed in the exercise. After a serious google search, the only reference I could find was a wiki article that claimed ordinal exponentiation was indeed $\Delta_1$. I couldn't make the same argument work here, since the corresponding order structure for $\alpha^\beta$ is the set $\{f:\beta\to \alpha\,|\, f\text{ has finite support}\}$ and I couldn't find an easy way to express this set in a $\Sigma_1$ way.

Is ordinal exponentiation $\Sigma_1$? If so, how does the proof go? (Hopefully I didn't miss an easy argument!)

(This also raised the question in my mind of whether or not, given $X$, the set of finite subsets of $X$ is $\Sigma_1$. I'd like to know the answer to that as well.)

Added: If I didn't make any mistakes, ordinal exponentiation is $\Delta_1$ if and only if that finite support set above is $\Delta_1$ definable. Thus, in principle, there should be an easy way to get a $\Sigma_1$ definition of that set. This seems surprising to me.