Proving $A\subseteq f^{-1}(f(A))$

Question

Given a function $f~:~X\to Y$ and a set $A\subseteq X$, we wish to prove the following:

$$A\subseteq f^{-1}(f(A))$$

My attempt:

Let $x\in A$. This implies $\exists y\in Y$ such that $f(x)=y$

Then $f^{-1}(f(x))=f^{-1}(y)=x$. Is this right?

Answer 1

If $a \in A$, then $f(a) \in f(A)$. Thus $ a \in f^{-1}(f(A))$.