Upon reviewing the basic theorem that the number of elements in a fixed finite set is unique, I tried to determine what part of this proposition is in need of proof. It seems axiomatic. Nonetheless, BBFSK have a very long-winded, and seemingly convoluted discussion of this, and related ideas.
When I attempted to produce my own argument in support of the above proposition, the part which I am not able to state purely in the terminology of mappings (bijection, injection, etc) is that an injection of a finite set into itself is a mapping onto itself. The proof BBFSK give uses induction. After thinking about it for a while, that was the only approach I could come up with.
Is there a rigorous proof of the proposition that every injective mapping of a finite set into itself is a mapping of the set onto itself which does not involve induction?