Isometry/Several characterizations with orthonormal basis/Fact

Let ${}V$ and ${}W$ be euclidean vector spaces, and let

$\varphi \colon V\longrightarrow W$

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

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

Write a proof