Prove that every $A \subset X$ satisfies $Y \smallsetminus f(A) \subset f(X\smallsetminus A)$

Prove that every $A \subset X$ satisfies $Y \smallsetminus f(A) \subset
f(X\smallsetminus A)$

Suppose that $f$ is surjective. I need to prove that every $A \subset X$
satisfies $Y \smallsetminus f(A) \subset f(X\smallsetminus A)$.
Here's what I have so far:
I suppose $y\in Y\smallsetminus f(A)$. Since $f$ is surjective, there
exists an $x \in X$ such that $f(x)=y$ by definition.
We also know that $y \not\in f(A)$. Is it possible at this point to say
that $x \not\in A$? I wasn't sure because there is no mention of $f$ being
invertible.
Let me know if I'm on the right track for this problem.