Understanding the Proof of Relative Openness Theorem in Rudin's Principles of Mathematical Analysis

I'm having trouble understanding Rudin's proof for the theorem stating:

"Suppose $Y \subseteq X$. A subset $E$ of $Y$ is open relative to $Y$ if and only if $E = Y \cap G$ for some open subset $G$ of $X$."

In particular, I'm having trouble understanding the direction going from right to left. He says:

"Conversely, if $G$ is open in $X$ and $E = G \cap Y$, every $p$ in $E$ has a neighborhood $V_p$ in $G$. Then $V_p \cap Y$ is a subset of $E$, so that $E$ is open relative to $Y$."

I'm having trouble understanding how to justify the last part where he concludes that the intersection $V_p \cap Y$ is a subset of $E$.