Invariant vector field of first integral

Question

I have to show that If $f:\delta\rightarrow\mathbb{R}$ is a first integral of $X$ a vector field then $M_{c}=f^{-1}(c)$ is invariant by $X$ I tried diferenciating a solution $\phi$ at an orbit and also looking at $M_{c}$ as a union of orbits
Edit: i don't know If It helps but the orbit $\phi_{t}(m)$ where $m \in M$ is restricted to $\delta$ therefore $X(\phi_{t}(m) \in \delta$

Edit2: the previous is wrong, aparently the way is looking at the tangent plane of an orbit over a Surface defined by $X$, but i'm strugling with what Surface is that, i also think my definition of invariant is not good

Answer 1

Apparently $M_c$ is invariant by $X$ If It is tangent to the field That translates to $df(\phi_{t}(d))* X(d)=0$ for every $d\in M_{c}$ But this comes from definition.
QED