The statement only holds for the one-dimensional case, the projective line.
One way to prove it is by showing that the cross ratio remains invariant under projectivities, and verifying that given three points, the cross ratio already uniquely defines a fourth.
Another approach would be by observing that a projectivity in $d$ dimensions is uniquely defined by $d+2$ points and their images, as a generalization of this post. So if $d+2$ points are fixed, the uniquely defined projectivity has to be the identity. In the general case you have to require the preimage and image points to be in generic position, e.g. no three collinear. For $d=1$ this reduces to a requirement for the points to be distinct.