an equivalent statement of "morphisms of projective varieties are closed"

Question

I am interested in seeing why the statement

(1) "If $Y$ is any variety and $Z$ a closed subset of $\mathbb{P}^n \times Y$, then the projection of $Z$ on $Y$ is closed."

implies the statement

(1) "A morphism of projective varieties takes closed sets to closed sets."

In particular, let $X,Y$ be projective varieties of $\mathbb{P}^m,\mathbb{P}^n$ respectively and let $f: X \rightarrow Y$ be a morphism. If the graph $G_f = \left\{ (x,f(x)), x \in X \right\} \subset \mathbb{P}^m \times Y$ was a closed subset of $\mathbb{P}^m \times Y$, then statement (1) would readily apply and $f(X)$ would be closed in $Y$ as desired. However, why need $G_f$ be closed in $\mathbb{P}^m \times Y$?

Reference: See Theorem 1 and above in Akhil Mathew's post here: http://amathew.wordpress.com/2010/10/23/a-projective-morphism-is-proper/#comments

PS: Thanks to Mariano for a correction.

Answer 1

I'm not sure what you have available to you, but it might be helpful to note that there's a map $f \times \operatorname{id}\colon X \times Y \to Y \times Y$, under which the inverse image of the diagonal is the graph of $f$. So if $Y$ is separated then the graph is closed.