Let $A$ be a $\mathbb Z_{\geq 0}$ graded ring. Then we have that $\operatorname{Proj}A$ is the set of homogenous prime ideals which do not contain the irrelevant ideal $A_+$. We put a topology on this by declaring $V(I)=\{\mathfrak p\in \operatorname{Proj}A:I\subset \mathfrak p\}$ to be the closed sets whenever $I$ is a set of homogenous elements of positive degree (equivalently let $I\subset A_+$ be a homogenous ideal of $A$). For any homogenous element $f\in A_+$, denote by $U_f$ the compliment of $V(f)$.
I am trying to show that the bijection: \begin{align} F:U_f&\longleftrightarrow \text{homogenous }\text{primes of }A_f\longleftrightarrow \operatorname{Spec}(A_f)_0 \end{align} given by: \begin{align} \mathfrak{p}&\longmapsto\mathfrak{p}_f\longmapsto \iota^{-1}(\mathfrak{p}_f) \end{align} is a homeomorphism. Here $\mathfrak{p}_f$ is the homogenous prime ideal of $A_f$ defined by: $$\mathfrak p_f=\left\{\frac{p}{f^k}:p\in \mathfrak p, k\geq0\right\}$$ and $\iota$ is the inclusion map $(A_f)_0\hookrightarrow A_f$. By $(A_f)_0$ I mean the degree zero elements of $A_f$, where $A_f$ is the $\mathbb Z$ graded ring with grading induced by the localization.
I first want to show that this is continuous, so let $I\subset (A_f)_0$ be an ideal, then I need to show that $F^{-1}(V(I))$ is a closed subset of $U_f$. We have that: \begin{align} F^{-1}(V(I))=&\{\mathfrak p\in U_f:\iota^{-1}(\mathfrak{p}_f)\in V(I)\}\\ =&\{\mathfrak p\in U_f:I\subset \iota^{-1}(\mathfrak p_f)\}\\ =&\{\mathfrak p\in U_f:\iota(I)\subset \mathfrak{p}_f\}\\ =&\{\mathfrak p\in U_f:\pi^{-1}(\iota(I))\subset \mathfrak p\} \end{align} where $\pi:A\rightarrow A_f$ is the localization map. However, my question is how can I be sure that $\pi^{-1}(\iota(I))$ contains only elements of positive degree. In particular, $\iota(I)=I$ as a subset of $A_f$ (in particular not an ideal), so there could be elements $a/1\in I$ where $a$ has degree zero in $A$, and then $\pi^{-1}(\iota(I))$ would contain elements that don't have positive degree, so I wouldn't be able to write this a homogenous ideal. Is there a reason why this couldn't happen?
Edit: So the definitions I am using are Vakhil, but looking at Liu's Algebraic geometry, for the vanishing locus, he lets $I$ be any homogenous ideal/homogenous subset. How could these be equivalent?