Isometric mapping of two subsets in a metric space

Let (M,d) be a metric space, and M1, M2 be two subspace of M.

Edit: I forgot to mension that both M1 and M2 are dense.

Suppose we have

$f: M rightarrow M$
and we know f is an isometric mapping from M1 to M1, i.e.,
$$
d(f(a),f(b))=d(a,b)
$$
for any $a,b in M1$.

Suppose we also have
$$
d(f(a),f(b))=d(a,b)
$$
for any $a in M1$ and $b in M2$.

The question is : can we say that f is an isometric mapping of $M1cup M2$?

In other words, is it possible to show that
$$
d(f(a),f(b))=d(a,b)
$$
for any $a,b in M1cup M2$.

I think it is true in Euclidean space, but I have no idea how to do it in a general metric space.

Any suggestion would be greatly appreciated.