Why do we have that $Bir(\mathbb{P}_{\mathbb{C}}^1)=Aut(\mathbb{P}_{\mathbb{C}}^1)$?

Question

I'm reading the notes where the following statement is assumed to be a well-known result:

Why is any birational map which maps the projective line $\mathbb{P}_{\mathbb{C}}^1$ onto itself belongs to $Aut(\mathbb{P}_{\mathbb{C}}^1)$?

I will appreciate any help!