Is This Set a Subset of the Other? Proof Assistance.

Question

If $f:X \to Y$ is a function and $U$ and $V$ are subsets of $X$, then $f(U\cap V) \subseteq f(U) \cap f(V) $.

I know that if this statement dealt with equality it would be false, but what about this? I have a feeling it is true since $\emptyset$ would be a subset of any set. Any proofs/guidance would be appreciated.

Answer 1

Hint:

$A\subseteq B\implies f(A)\subseteq f(B)$.