I encountered this theorem on Wiki "there does not exist an injective function whose codomain is smaller than its domain". Here is my attempt to prove it, can you please have a look if this proof is OK?
Proof: Let's say we have: $f:A\to B$ where $|A| = N$ and $|B| = M$. Let's take all elements from $A$:
$ a_1,\cdots ,a_N $ and by mapping them, we get:
$f(a_1),\cdots , f(a_N)$.
There can't exist two same elements in latter list (according to injection). Which means there need to be at least N distinct elements in B. This can't hold if $M < N$.