Isometry/Several characterizations with orthonormal basis/Fact/Proof/Exercise

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

$\phi :V⟶W$

denote a linear mapping. Show that the following statements are equivalent.

1. $\phi$ is an isometry.
2. For every orthonormal basis $u_i$, $i=1,\dots ,n$, of $V$, $\phi (u_i)$, $i=1,\dots ,n$, is part of an orthonormal basis of $W$.
3. There exists an orthonormal basis $u_i$, $i=1,\dots ,n$, of $V$ such that $\phi (u_i)$, $i=1,\dots ,n$, is part of an orthonormal basis of $W$.

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