For the ring of $p$-adic integers $\mathbb Z_p$ let $\mu_n: \mathbb Z_p \to \mathbb Z / p^n \mathbb Z$ be the projection mapping. Consider $\mathbb Z / p^n \mathbb Z$ with the discrete topology.
Is (for $v \in \mathbb Z / p^n \mathbb Z$ and $w \in \mathbb Z_p$ with $\mu_n(w)=v$) the following equality of sets $$\left \{ y \in \mathbb Z_p : \mu_n(y)=v \right \} = \left \{ y \in \mathbb Z_p : v_p(y-w)>n \right \}$$ correct and if yes, why?
(Here, $v_p(.)$ is the discrete valuation on $\mathbb Z_p$.)
There are many definitions of $\mathbb{Z}_p$, I assume that you take it to mean
$$\varprojlim\mathbb{Z}/p^m\mathbb{Z}=\left\{(a_m)\in\prod_m\mathbb{Z}/ p^m\mathbb{Z}:a_{m+1}\equiv a_m\mod p^m\right\}$$
The projection map is then given by $\mu_n((a_m))=a_n$. Moreover,
$$v_p((a_m))=\min\{m:a_m\ne 0\}$$
So now, we see that $\mu_n((y_m))=v$ is actually equivalent $y_i\equiv v\mod p^i$ for $i=1,\ldots,n$. Which, is equivalent to, by what we've just said, $w_i\equiv y_i$ for $i=1,\ldots,n$, which is equivalent to $v_p((w_m)-(y_m))>n$.