Isometry/Several characterizations with orthonormal basis/Fact

Let $V$ and $W$ be Euclidean vector spaces, and let

φ : V ⟶ W

denote a linear mapping. Then the following statements are equivalent.

$φ$ is an isometry.
For every orthonormal basis $u_{i}$ , $i = 1, \dots, n$ , of $V$ , $φ (u_{i})$ , $i = 1, \dots, n$ , is part of an orthonormal basis of $W$ .
There exists an orthonormal basis $u_{i}$ , $i = 1, \dots, n$ , of $V$ such that $φ (u_{i})$ , $i = 1, \dots, n$ , is part of an orthonormal basis of $W$ .