If you're asking for a proof that "rolling without slipping" means $v=r\omega$, here's one:
When we say an object is "rolling without slipping," that means that the distance covered by the object in a certain time $\Delta t$ is exactly the same as the arc-length that a point on the wheel travels while rotating. (To see why this is true, consider the opposite: if a wheel is slipping, then one of two things are happening: it either travels some distance without turning, or turns without traveling any distance. In the former case, the arc-length that a point on the wheel travels is less than the distance covered; in the latter case, it is greater. Requiring that it doesn't slip therefore means that neither of the above are true, guaranteeing equality. You can also see this if you imagine "rolling up" the ground onto a wheel as it rolls, or equivalently, if you imagine the wheel as a spool of string that unravels as it rolls.)
In a time interval $\Delta t$, a wheel rotating with constant angular velocity $\omega$ will rotate a total arc-length of $s=r\theta=r\omega\Delta t$. Similarly, in a time interval $\Delta t$, an object traveling at constant velocity $v$ will cover a distance $d=v\Delta t$. Setting $s=d$ as per our argument in the first paragraph, we see that
$$v\Delta t=r\omega\Delta t$$
Since we have imposed no other assumptions besides "rolling without slipping" in the above, we have therefore proven that saying an object is rolling without slipping is equivalent to setting $v=r\omega$.