My question is related to this question
I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is a $C^r$ submanifold of $M$.
I know that I need to show that $f$ has constant rank in a neighbourhood $U$ (open in $M$) of $A$ so I can apply the Rank Theorem to finish the proof. I tried the same approach as in P. W. Michor, Topics in Differential Geometry, section 1.15, but since the hypothesis in this book is kinda different I couldn't use it to get the result I need.
I thought that since $f\vert_A=id:A\rightarrow A$, then for every $x\in A$, $T_xf=Id$ which is surjective. Therefore $f$ is a local submersion in $x$ and so, in a neighbourhood $U_x\underset{open}\subset M$ $f$ has constant rank. Now I can collect all $U_x$ and get an open set $U$ in $M$ where $f$ has constant rank.
Is this correct?
By the way, this is question 2, Section 2 Chapter 1 from Hirsch's "Differential Topology".
Thanks.