Consider the following statement of the inverse function theorem:
Let $f : W \to \mathbb{R}^n$ be a $C^1$ function from an open subset $W \subseteq \mathbb R^n$ and let $p \in W$. $f$ has a local $C^1$ inverse if $Df(p)$ is invertible.
This is effectively what one sees in Rudin's Principals of Mathematical Analysis. The question is: can the "if" can be replaced by "iff"? It seems like the answer is obviously "yes": just apply the chain rule to $f \circ g = 1$ and $g \circ f = 1$. Despite this, I can't find a statement that has the inverse function theorem in the "iff" form, so I feel like I must be making some kind of dumb mistake.
For example this question has someone saying it only holds for "low dimensions" and someone else saying that it suffices to require the inverse to be surjective, which doesn't make sense to me given the straightforward proof I gave above.
What am I missing?