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I am currently doing an exercise and I want to show my solution so far and ask for an hint about the last part.

Exercise: Let $M$ and $N$ be smooth manifolds and let $S \subset M \times N$ be a submanifold. Denote by $\pi_M : M \times N \rightarrow M$ a projection and similar for $\pi_N : M \times N \rightarrow N$. Show that the following are equivalent:

(a) $S$ is the graph of a smooth map $\phi : M \rightarrow N$;

(b) $\pi_M\mid_S$ is a diffeomorphism from $S$ onto $M$;

(c) For each $p \in M$, the submanifolds $S$ and $\{p\} \times N = \pi_M^{-1}(p)$ intersects transversely and the intersection consists of a single point.

Attempt:

$(a) \implies (b)$. Since the projection is a smooth map and the restriction of smooth maps to submanifolds is smooth, $\pi_M\mid_S$ is the restriction of the smooth map $\pi_M$ to the submanifold $S$. Hence smooth. The inverse is given by $p \mapsto (p,\phi(p))$ which is smooth by (a).

$(b) \implies (c)$. The fact that $\pi_M\mid_S$ is diffeomorphic to $M$, gives us the fact that $d_{(p,q)}\pi_M\mid_S : T_{(p,q)}S \rightarrow T_pM$ is a diffeomorphism. This means that $\dim(T_{(p,q)}S) = \dim(T_pM) = m$. Furthermore, $\{p\} \times N$ is also diffeomorphic with $N$ via projection, so that $\dim(T_{(p,q)}\{p\}\times N) = \dim(T_qN) = n$. Therefore $$\dim(T_{(p,q)}S) + \dim(T_{(p,q)}\{p\}\times N) = m + n = \dim(T_{(p,q)}(M \times N)),$$hence they intersect transversely. The fact that the intersection is in a single point is due to the fact that $S \cap \{p\} \times N = \{\pi_M\mid_S^{-1}(p)\}$.

$(c) \implies (a)$. We define $S = \{(p,q) : q = \pi_M\mid_S^{-1}(p)\}$ which is well defined because the intersection is in a single point. Now it remains to show that $\pi_M\mid_S^{-1}$ is in fact smooth.

Question: I wonder if my reasoning is correct and what is the missing piece in order to show that $S$ is the graph of a smooth function.

Thank you in advance!

Edit: I saw this post but they use (c) to prove (b). I am trying to show that it implies (a).

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