An element in a ring reduced modulo a prime ideal

Question

I am reading scheme theory and this thing in the prime spectrum perplexed me. An element in a ring $A$ reduced modulo a prime ideal $\mathfrak{p}$ is an element in the residue field of $A_{\mathfrak{p}}$. For example, $A = \mathbb{Z}$. Then $25\ mod\ (17)$ is $\overline{8}$ in field $\mathbb{F}_{17}$. How do we prove that the reduced element is in the residue field rather than just an element in $A/(\mathfrak{p})$?