Let $X$ be a connected $n$-dimensional differentiable manifold, and $f: X \rightarrow X$ a differentiable map such that $f \circ f = f$. Now I have to show that the image $f(X)$ is a submanifold of $X$, using the following sub-results:
$\mathrm{rk}_p f \leq \mathrm{rk}_{f(p)} f$ for all $p \in X$
The rank of $f$ is constant along $f(X)$
The rank of $f$ is constant in an open neighbourhood of $f(X)$
I have managed to show that $f(X)$ is a submanifold when the above holds (using the constant rank theorem), but I don't see how to prove the above results.