# Unique projective maps preserving lines

Let $$\mathbb{P}(V)$$ be a projective space of dimension $$3$$ and let $$L_i$$, $$i=1,2,3$$ be pairwise non-intersecting lines in $$\mathbb{P}(V)$$. If $$\phi: L_1 \to L_1$$ is a projective transformation, prove that there is a unique projective transformation $$\Phi: \mathbb{P}(V) \to \mathbb{P}(V)$$ such that $$\Phi(L_i) = L_i$$ and $$\Phi_{|L_1} = \phi$$.

It might be helpful to use that if $$L_i = \mathbb{P}(U_i)$$ then $$U_i \oplus U_j = V$$ and hence that there are unique maps $$\alpha : L_1 \to L_2$$ and $$\beta: L_1 \to L_3$$ such that $$x$$, $$\alpha(x)$$, $$\beta(x)$$ are collinear for all $$x \in L_1$$. (I was able to prove this, but cannot apply it properly).

Any help appreciated!