analysis - Understanding the definition of surjective function

From Wikipedia, I've seen that the definition of surjective function is the following:

If
$$ f \colon X \rightarrow Y$$ then, $f$ is said to be surjective if
$$ \forall y \in Y\, \exists x \in X\, \text{s.t. } f(x) = y \text{.}$$

On the other hand, we can define surjectiveness by saying that $f$ is surjective iff the codomain of $f$ is equal to its image, $f(X)$, i.e. if:

$$ \text{codomain of } f \text{ } = f(X) = \{y \colon y \in Y \wedge (\exists x \in X \colon f(x)= y)\} \text{.} $$

Why these two definition are the same? And, in general, how can i prove it?

Answer

In general the image of a function $f\colon X \rightarrow Y$ is denoted with $f(X)$, and by definition it is a subset of $Y$, so $f(X) \subseteq Y$. If $f$ is surjective, then $Y \subseteq f(X)$, because for every $y \in Y$ you can find $x \in X$ such that $y = f(x)$, so $y \in f(X)$. To sum up, $f(X) \subseteq Y$ and $Y \subseteq f(X)$, which by double inclusion means $f(X) = Y$.