Let $p$ be an odd prime. Let $(1+p\mathbb{Z}_p)$ denote the multiplicative group of p-adic numbers 1 mod p. Why is $$\mathbb{Z}_p[[(1+p\mathbb{Z}_p)]] \cong \mathbb{Z}_p[[t]] $$ (non-canonically) via $(1+p) \mapsto 1+t$?
This is claimed many places but I never see the details anywhere I read.
By definition we have inverse limit $$\mathbb{Z}_p[[(1+p\mathbb{Z}_p)]] = \lim \mathbb{Z}_p[(1+p\mathbb{Z}_p)/(1+ p\mathbb{Z}_p)^n]$$ where on the right hand side we have the group algebra. I'm struggling to understand for sure what elements of $(1+p\mathbb{Z}_p)/(1+ p\mathbb{Z}_p)^n$.