Composing dominant rational map from an irreducible scheme $X$ to an irreducible scheme $Y$ with rational map from $Y$ to $Z$

Question

I am trying to understand a claim Vakil makes in passing in FOAG between 6.5.A and 6.5.B - he claims what I stated in the subject line. He defines a rational map as dominant if for some representative the image is dense. What I am getting stuck with is why, if you have $\phi: U \to Y$ and $\rho: V \to Z$, where $U,V$ are dense, you can find some dense open subset of $U$ that gets mapped to $V$. I tried to show this as a general topological fact, i.e. that pre-image of a dense open set is open and dense under a continuous map if the image of the continuous map is dense in the target space, but I did not succeed. Appreciate any hints.