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Consider the subsetIt is nonempty by the given initial data. A local isometry preserves the Levi-Civita connection and therefore affinely parametrized geodesics. On a sufficiently small normal neighborhood of $x\in A$, this givesThe right sides agree, so $f_1=f_2$ on that neighborhood and their differentials agree there as well. Consequently $A$ is open.

It is also closed. If $x_j\in A$ converges to $x$, continuity first gives $f_1(x)=f_2(x)$. Choose coordinate charts around $x$ and this common image. The coordinate matrices of the two differentials depend continuously on the base point; their equalities at $x_j$ therefore pass to $x$. Thus $x\in A$. Since $M$ is connected, the nonempty subset $A$ that is both open and closed equals $M$. Hence $f_1=f_2$ everywhere. This proves local isometries are determined by first-order data without requiring completeness.