
It looks like you're working on the setup of the problem. I think I would word it like this:
$\displaystyle U\subset X$; therefore, $\displaystyle (\forall u)(u\in U\to u\in X).$ Now $\displaystyle \phi:X\to Y.$ Therefore, $\displaystyle \phi(X)\subseteq Y,$ and hence $\displaystyle \phi(U)\subseteq Y.$ It follows that $\displaystyle (\forall u)(u\in U\to\phi(u)\in Y).$
It's a good thing, by the way, that you're concerned about good ways of writing things. Far too few students are concerned about that! And those students who don't care about careful and clear wording are getting Ph.D.'s, and writing incomprehensible dissertations and even more incomprehensible research papers. And we wonder why more people don't want to go into math and science! Communication is essential  I'm sure you already know that. If you're interested in learning more about scientific writing, one book I'd highly recommend is N. David Mermin's Boojums All the Way Through. Love that book.