Reference request for an elementary identity in cardinal arithmetic


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…)