If we have a transformation from $(X,Y) \to (W,Z)$, why is it one-to-one if and only if we can solve for $(X,Y)$ in terms of $(W,Z)$?

Question

I read somewhere that a transformation of random variables $(X,Y) \to (W,Z)$ is one-to-one if and only if we can solve for $(X,Y)$ in terms of $(W,Z)$. However, I am not seeing this intuitively. It seems that the only condition of being one-to-one is that every element in the range has only one corresponding value that maps to it in the domain. Can someone give an intuition here? thanks