Why do two vector fields determine a paramaterisation of a surface

I am trying to understand the following proof about how two vector fields determine a paramaterisation of a surface. What I do not understand is why the inverse of the constructed $$\phi$$ is the parametrisation we are looking for. I tried to differentiate the parameter curve $$\phi^{-1}(t,p_{2})$$, but don't see why the resulting vector is parallel/linear dependend to/of $$w_{1}$$.

The proof is from DoCarmos book Differential Geometry of Curves and Surfaces on page 185.

• $\varphi^{-1}(t,p_2)$ gives points $q$ with $f_2(q)=p_2$, and by construction the level curves of $f_2$ are tangent to $w_1$. You have to go back and look at what the first integral of a vector field is. For an alternative proof, see pp. 119-120 of my differential geometry text. – Ted Shifrin Oct 12 '18 at 19:11