I don't know what you think is "misleading" about "mapping/relation theory". For that matter, I don't know what you mean by "mapping/relation theory" that is not just part of basic set theory.
In fact you don't even need to do that. In fact I prefer to use just X or something so as to avoid the misleading language of mapping / relation theory.
See what I did in:
Exists Bijection to a Disjoint Set - ProofWiki