Let $X$ and $Y$ be sets with $m$ and $n$ elements respectively and let $f:X\rightarrow Y$ be a function from $X$ to $Y$.
Show that if $f$ is surjective, then $m\geq n$.
I know this should be true since in words a surjective can be seen as:
"For every element in the codomain, there is at least one element of the domain mapping onto it".
I'm having trouble converting this idea into a proof. My guess is I need to show that at least $n$ maps from $Y$ to $X$ are possible for distinct values of $x$. If this is the case, we can assert $m \geq n$.
So far I basically have the definition with quantifiers.
Proof: Show than at least $n$ distinct maps from the codomian, back to the domain are possible.
Since $f$ is surjective, $\forall y \in Y, \exists x \in X : f(x) = y.$