Does anyone know if these following two statements can be proved somehow or are they simply true by definition?

"If there exists a surjective function $\displaystyle f: X \to Y$, then $\displaystyle card(X) \geq card(Y)$"

"If there exists an injective function $\displaystyle f: X \to Y$, then $\displaystyle card(X) \leq card(Y)$"

Both statements are invoked in more advanced proofs without hesitation, so I usually take them for granted. But I realized that's a bad mentality, so I tried to prove those two statements using the definition of injection and surjection but I am not sure if my approach was correct.

For a surjection, it specifies that for all $\displaystyle y \in Y$, there exists an $\displaystyle x \in X$ such that $\displaystyle f(x) = y$. The definition never specified that such an $\displaystyle x$ was unique, so we could construct a mapping of two elements or more elements in $\displaystyle X$ towards one element in $\displaystyle Y$, which would get us a comparison of the cardinality of the two sets. Is that correct (and if it is, rigorous enough)?

For an injection, it only says that if two elements $\displaystyle f(x_1) = f(x_2) \in Y$, then $\displaystyle x_1 = x_2$. It never talked about such an element $\displaystyle y \in Y$ which has no corresponding $\displaystyle x$ element to map from, which would suggest that $\displaystyle card(X)$ could be less than $\displaystyle card(Y)$. Would that work?

I could not find anything online that delved further into the matter. Could someone please shed some light on this matter?

Thanks!