Let $\kappa$ be an infinite cardinal. Then we have
$$\beth_{\kappa+1} = \prod_{\alpha<\kappa} \beth_{\alpha} = \beth_{\kappa}^{\kappa}$$
The only non-trivial inequality is the first less-than-or-equal-to which can be obtained by coding each $X \in \mathcal{P}(\beth_{\kappa}) \cong \beth_{\kappa+1}$ as the map $f \in \prod_{\alpha<\kappa} \beth_{\alpha}$ defined by $f(\alpha)=g_{\alpha}(X \cap \beth_{\alpha-1})$ for non-zero non-limit $\alpha$, where $g_{\alpha}: \mathcal{P}(\beth_{\alpha-1})\rightarrow \beth_{\alpha}$ is a fixed bijection, and $f(\alpha)=0$ for other $\alpha$.
I need to use this identity in a manuscript (intended for non-set theorists) and want to avoid including a full proof to save space. Does anyone know of any textbook which contains this as a fact? The closest one I was able to find is Exercise I.13.33 in Kunen's (new) book which only covers the case $\kappa=\omega$. (Worst case, I'll write down the proof myself but given Kunen's exercise, this product shouldn't be that unusual and must have been written down somewhere...)