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Let $f:X\to Y$ be one-to-one and onto (i.e., bijective). Prove that $f(X)=Y$, where $f(X)=\{y:y=f(x)$ for some $x\in X\}$

I have seen this result stated a few times, such as here: Co-domain & Image. But I was struggling to write a formal proof from the definition of onto and one-to-one. I have managed to show that since $f$ is onto, $y\in Y\implies y\in f(X)$, so $Y\subset f(X)$, but I'm not sure about how to show $f(x)\subset Y$. Any help would be appreciated!

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    $\begingroup$ $f(X)\subset Y$ because $f$ is defined to be a map from $X$ to $Y$, i.e. every value of $f(x)$ is in $Y$. $\endgroup$ – helloworld112358 May 16 '17 at 5:06
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$f(x) \in Y,$ since $f$ is a map from $X$ to $Y.$

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