Let $F$ be a (class) function that assigns at each cardinal $\alpha$ the minimum cardinal $\beta$ such that $\alpha < \alpha ^ \beta$. I have to prove that $\beta \in \text{Im}(F)$ iff $\beta$ is regular (i.e. $\text{cof}(\beta) = \beta)$.
I proved the "only if" part, in fact if $\beta = F(\alpha)$ is not regular then let $\mu < \beta$ be it's cofinality. This means we can write $\beta = \sum_{i \in \mu} \beta_i$, with $\beta_i <\beta$ for each $i \in \mu$. So we have: $$\alpha <\alpha^\beta = \alpha ^ {\sum_{i \in \mu} \beta_i} = \prod_{i \in \mu} \alpha^{\beta_i} = \prod_{i \in \mu} \alpha = \alpha^\mu$$ Where the third equivalence is due to the fact that $\beta$ is the minimum s.t. $\alpha < \alpha^\beta$, this is absurd because $\mu<\beta$ but $\alpha < \alpha^\mu$.
What about the "if" part?