elementary set theory - Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$

I haven't been able to do this exercise:

Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse
image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \subseteq B$.

Since $X \subseteq B$, all the elements in $X$ belong to the codomain of $f$.

Since $f$ is surjective, it means that all elements in the codomain $B$ have some preimage in $A$. Given that $X \subseteq B$, all elements in $X$ must also have a preimage in $A$.

Have $\triangle = f^{-1}(X)$, $\triangle$ is now a set containing the preimages of the elements in $X$. Because of this, $\triangle \subseteq A$.

If we evaluate $f(\triangle)$, we...... nope, I don't know what I'm doing now.

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What do you think?