Find the MLE of $e^{-\theta}$

Question

Let $X_{i} \sim Poisson(\theta)$ be independent random variables. I know that the MLE of $\theta$ is the sample mean which is $\frac{\sum X_{i}}{n}$. The question asks me to find the MLE of $e^{-\theta}$. I suspect that it is to be $e^{- \overline X} = e^{- \frac{\sum X_{i}}{n}}$. But how can I determine that Maximum likelihood estimator of $e^{-\theta}$?