# Show that the image of the domain of a bijective function is the co-domain of the function

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!

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

$f(x) \in Y,$ since $f$ is a map from $X$ to $Y.$