elementary set theory - Need help for proving that: $f(f^{-1}(A)) ⊆ A$

I realised I needed to show more information, which I now did:

$$f: X \rightarrow X \,\,\,\rm{and}\,\,\,\, A \subseteq X$$

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

This is my proof:

By defintion:

$$f^{-1}(A)=\{x \in X\mid f(x) \in A\}$$
and

$$f(A)=\{f(x) \mid x \in A\} = \{y \in X \mid \exists x \in A: y=f(x)\} \subseteq X$$

Therefore we can end the proof by a final definition:\
$$f(f^{-1}(A))=\{y \in A: \exists x \in f^{-1}(A):y=f(x)\} \subseteq A$$

Is this a legit "proof"? And is it even a proof, when i only use definitions?