I suffer from similar habbits. Always assuming things are complicated. Your proof is correct in its simplicity. You are absolutely right to assume the existence of , its given in the problem statement. Hence you may use the identity function , by the definition of the inverse.
In fact just to reassure you that your proof makes sense, remember that f is the inverse of the function . And a function only has an inverse if it is strictly increasing or decreasing (a standard theorem from set theory). You have of course proved is strictly increasing. And thus the inverse of exists. namely, f.