 Singular strong limit power-set identity

For a singular strong limit cardinal $\lambda$, let $\mu =\mathrm{cf}(\lambda )$ and choose cofinal cardinals ${\lambda }_i<\lambda$. Subset restrictions inject $\mathcal{P}(\lambda )$ into ${\prod }_{i<\mu }\mathcal{P}({\lambda }_i)$, whose cardinal is at most ${\lambda }^{\mu }$ by the strong-limit property. Conversely ${\lambda }^{\mu }\le (2^{\lambda }{)}^{\mu }=2^{\lambda }$. This equality alone does not imply the singular cardinals hypothesis.