Let $\varphi:A\to B$ be a ringhomomorphism. Show that the induced map $$^a\varphi:Spec(B)\to Spec(A),\ \mathfrak{p}\mapsto \varphi^{-1}(\mathfrak{p})$$ satisfies $$\overline{^a\varphi(V(\mathfrak{b}))} = V(\varphi^{-1}(\mathfrak{b}))$$ for any ideal $\mathfrak{b}$ in $B$. Here, for a ring $A$ we have defined $V(S)=\{\mathfrak{p}\in Spec(A)|S\subseteq p\}$ (these are the closed sets of the Zariski topology on $Spec(A)$).
I have already shown in some earlier exercise that $^a\varphi$ is well-defined and continuous. Further I was able to show that $^a\varphi(V(\mathfrak{b})) \subseteq V(\varphi^{-1}(\mathfrak{b}))$, and since $V(\varphi^{-1}(\mathfrak{b}))$ is closed, it only remains to show that $$V(\varphi^{-1}(\mathfrak{b})) \subseteq W$$ for any closed $W$ containing $^a\varphi(V(\mathfrak{b}))$. I am stuck in here, any help would be appreciated.