# homomorphism / commutative property

$U\subset X$; therefore, $(\forall u)(u\in U\to u\in X).$ Now $\phi:X\to Y.$ Therefore, $\phi(X)\subseteq Y,$ and hence $\phi(U)\subseteq Y.$ It follows that $(\forall u)(u\in U\to\phi(u)\in Y).$