Let $f:M^{m}\rightarrow N^{n}$ a differential application and $S^{k}$ a submanifold of $N$. We say that $f$ is transversal to $S$ in $p\in f^{-1}(S)$ if $df_{p}(T_{p}M)+T_{f(p)}S= T_{f(p)}N$. If $S^{0}=\{c\}$, show that $f$ is transversal to $c$ if and only if $c$ is a regular value of $f$.
My work:
Suppose that $f$ is transversal to $c$, so: \begin{eqnarray} df_{p}(T_{p}M)+T_{f(p)}S= T_{f(p)}N \end{eqnarray} if $v\in T_{f(p)}N$, then: \begin{eqnarray} v=\lambda_{i}\sum_{i} df_{p}\left(\frac{\partial}{\partial x_{i}}\right)+\beta_{j}\sum_{j} \frac{\partial}{\partial u_{j}} \end{eqnarray} where $T_{f(p)}S=span\left\{\displaystyle\frac{\partial}{\partial u_{j}}\right\}$ and $T_{p}M= span\left\{\displaystyle\frac{\partial}{\partial x_{i}}\right\}$, so, every $v\in T_{f(p)}N$ can be write in terms of $df_{p}$ and $df_{p}:T_{p}M\rightarrow T_{f(p)}N$ is a surjection (this conclusions isn't clear for my right now).
By other hand, Suppose that $c$ is a regular value of $f$, so: \begin{eqnarray} \forall p\in f^{-1}(c), \hspace{0.5cm} df_{p}:T_{p}M\rightarrow T_{f(p)}N \end{eqnarray} is a surjection between $T_{p}M$ and $T_{f(p)}N$, so, analogous to before case, $\forall v\in T_{f(p)}N$: \begin{eqnarray} v&=&\lambda_{i}\sum_{i}df_{p}\left(\frac{\partial}{\partial x_{i}}\right)\\ \end{eqnarray} but I can't express it in span's terms of $df_{p}$ and $T_{f(p)}S$. I'll appreciate advices or ideas to continue in both cases.