In Abbott's Understanding Analysis, he defines cardinality as follows:
The set $A$ has the same cardinality as $B$ if there exists $f : A → B$ that is 1–1 and onto. In this case, we write $A ∼ B$.
Do we need a function to exist? I mean, can't you just say sets A and B are 1-1 and onto? I know that the definitions of 1-1 and onto describe functions, not sets. BUT, later he proves that $\mathbb{Q}$ is countable and it just seems like an awful lot of trouble to provide a function for that example. And whether or not you can provide an explicit function seems like a moot point in deciding whether or not two sets have the same cardinality.
It isn't that we need a function to exist; rather, a bijection must exist because the sets are one-to-one and onto. The primary problem with just saying the cardinalities are the same is that it's hard to directly compare cardinalities for infinite sets, which is why our function definition works --- it allows us to put two sets into correspondence with each other.