Proposition 9.1.7.12. Assume the generalized continuum hypothesis, and let $\kappa < \lambda $ be regular cardinals. Then $\kappa \triangleleft \lambda $ if and only if $\lambda $ has exponential cofinality $\geq \kappa $.
Proof. Assume that $\kappa \triangleleft \lambda $; we will show that $\lambda $ has exponential cofinality $\geq \kappa $ (the converse follows from Proposition 9.1.7.8). If $\lambda $ is strongly inaccessible, this is immediate. Otherwise, we may assume that $\lambda = \lambda _0^{+}$ is a successor cardinal (Remark 4.7.3.23). Let $\{ S_ i \} _{i \in I}$ be a $\kappa $-small collection of $\lambda $-small sets; we wish to show that the product $\prod _{i \in I} S_ i$ is $\lambda $-small. Fix a set $S$ of cardinality $\lambda _0$. Each of the sets $S_ i$ has cardinality $\leq \lambda _0$, and is therefore isomorphic to a subset of $S$. We may therefore assume without loss of generality that each $S_ i$ is equal to $S$, so that $\prod _{i \in I} S_ i$ can be identified with the set $\operatorname{Fun}(I, S)$ of functions from $I$ to $S$. Set $T = I \times S$. Note that every function $f: I \rightarrow S$ is determined by its graph $\Gamma _{f} = \{ (i,s) \in I \times S: f(i) = s \} $, which is a $\kappa $-small subset of $T$. We will complete the proof by showing that the collection of $\kappa $-small subsets of $T$ is $\lambda $-small.
Our assumption that $\kappa \triangleleft \lambda $ guarantees that there is a $\lambda $-small collection $\{ T_ j \} _{j \in J}$, where each $T_ j$ is a $\kappa $-small subset of $T$ and every $\kappa $-small subset of $T_ j$ is contained in some $T_ j$. It will therefore suffice to show that, for each $j \in J$, the collection of subsets of $T_{j}$ is $\lambda $-small. This is clear: since $T_ j$ has cardinality $< \kappa $, the generalized continuum hypothesis guarantees that the collection of subsets of $T_ j$ has cardinality $\leq \kappa < \lambda $. $\square$