I am reading this note of Blickle on motivic integration and I am stuck at a technical point. At the beginning of the appendix, we are given a proper birational morphism $f: X' \longrightarrow X$ between smooth $k$-varieties ($k$: a field). In particular, we have a first fundamental sequence $$f^*\Omega^1_{X/k} \overset{df}{\longrightarrow} \Omega^1_{X'/k} \longrightarrow \Omega^1_{X'/X} \longrightarrow 0.$$ It is claimed that this sequence is also exact on the left by the birationality of $f$. It is hard to believe this as birationality just implies that $X$ and $X'$ are isomorphic over some open set but not on the whole spaces.
I tried to google, and in the same situation as the OP of this topic, I find no reference for the sequence in the case of blowing up. Assume that it holds true for blowing up, then can we deduce this from the weak factorization theorem (theorem 3.6, page 19 in Blickle)?
Both $f^*\Omega_X$ and $\Omega_{X'}$ are locally free sheaves (because $X$ and $X'$ are smooth) and the morphism $df$ is an isomorphism on a dense open subset open (by birationality of $f$). Now the kernel of $df$ is a subsheaf of a locally free sheaf, hence it is torsion free, and it is zero on a dense open subset, hence it is torsion. A combination of these two observations implies it is zero, hence the sequence is left exact.