Range of a P-name I am working on a set theory problem from Kunen's Set Theory book, and it involves knowing $\text{ran}(\tau)$ where $\tau$ is a $\mathbb{P}$-name.  The entire section loves to talk about the domain of things like $\tau$, but not the range.  Is 
$$\text{ran}(\tau) = \{p \in \mathbb{P} : \exists \sigma \in M^{\mathbb{P}}(\langle \sigma, p \rangle \in \tau)\}$$ correct?  I don't think I have the right thing since this doesn't seem to help me solve my problem.

The name $\tau\in M^\mathbb{P}$ is an element of the set $M$, and in particular, it is a relation. So you can compute its range 
$$
\mathrm{ran}(\tau) = \{p\in \mathbb{P} : \exists \sigma\,(\langle\sigma,p\rangle\in\tau)\}.
$$
But in this particular problem, $\tau$ is the name of a function and you have the expression
$$
p\Vdash \check b\in\mathrm{ran}(\tau).
$$
So you have to read the expression to the right of $\Vdash$ as a formula of the forcing language. Therefore, $p$ forces that $b$ belong to the range of the function denoted by $\tau$. In other words, using the Fundamental Theorem,
$$
M[G]\models b\in \mathrm{ran}(\tau_G)
$$
for every generic such that $p\in G$ and $b\in M$.
