Establish isometries between $\mathcal{B}(X;M\times N)$ and $\mathcal{B}(X;M)\times \mathcal{B}(X;N)$

Question

Establish isometries between $\mathcal{B}(X;M\times N)$ and $\mathcal{B}(X;M)\times \mathcal{B}(X;N)$, where $X$ is an arbitrary set, $M,N$ are metric spaces and $\mathcal{B}(X,M)$ represented the set of bounded functions.

I need define a function such that $F:\mathcal{B}(X;M\times N)\to\mathcal{B}(X;M)\times \mathcal{B}(X;N)$. And if we take $f,g\in\mathcal{B}(X,M)$, then the distances $d(f,g)=\displaystyle\sup_{x\in X}d(f(x),g(x))<\infty$.

Answer 1

It seems that putting $F(f)(x)=(\pi_M f(x), \pi_N f(x) )$ for any function $f\in\mathcal{B}(X;M\times N)$ and any $x\in X$, where $\pi_M$ and $\pi_N$ are the projections of the product $M\times N$ onto its factors, is OK.