# The direct image of the inverse image of a surjective map

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

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Im not an expert or teacher in elementary set theory but from my experience i recall learning your definions well pays off well in trying to prove a short statement. I noticed that Asaf gave a correct answer in the meanwhile and i note that taking examples such as " let x be element of " occurs very often in elementary set theory proofs as well as " pulling an = into 2 times a C " if you know what i mean. You must master those skills for the future. – mick Sep 2 '12 at 21:38

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 $x\in H$.