The direct image of the inverse image of a surjective map

Question

If $f:A\rightarrow B$ is a surjection and $H\subseteq B$, how to show that $$ f(f^{-1}(H))=H $$

Answer 1

Let $x\in H$, since $f$ is surjective there is some $a\in A$ such that $f(a)=x$, and this $a\in f^{-1}(H)$ by definition (since $x\in H$). Therefore $f(a)=x\in f(f^{-1}(H))$. Therefore $H\subseteq f(f^{-1}(H))$.

I leave you to do the second direction, pick some $x\in f(f^{-1}(H))$ and use the definition of $f(Y)$ and $f^{-1}(Z)$ for $Y\subseteq A, Z\subseteq B$, to conclude that $x\in H$.