how to see that isometries are distance preserving

Question

A riemannian metric induces a distance function that makes the riemannian manifold a metric space. Intuitively, it certainly should be true that every isometry is distance-preserving. But how to actually see this? Via pull-back of the metric, the speed of the (piece-wise) curve can be transformed under an isometry, but how to see that the infimum of lengths of transformed curves is the same as the original? Thanks!