What is the difference between $\pi \in \sigma$ to $\pi \in dom(\sigma)$ I am reading the proof of 4.20 here below and I don't understand:
What is the different between $\pi \in \sigma$ to $\pi \in dom(\sigma)$?
I am also not sure what is $dom(\sigma)$. Is there anything in the Lemma that allows me to state that $\sigma$ is a function or a relation? Because if not then what can $dom(\sigma)$ other the domain of a relation or a function?
Thank you! 
 
 A: $\sigma$ must be a $\mathbb P$-name.
This means that in the forcing language, that is, the formulas we write after $\Vdash$, "$\sigma$" counts as a constant. Eventually it will name an individual in the model $M[G]$, so $\sigma$ can also appear as a constant to the right of "$M[G]\vDash$".
However, within $M$ where we say "there is some $\pi\in\operatorname{dom}(\sigma)$", a $\mathbb P$-name is a set of pairs $\langle \tau,p\rangle$ where $\tau$ is itself a $\mathbb P$-name, and $p\in\mathbb P$. What $\pi\in\operatorname{dom}(\sigma)$ must mean is then that $\pi$ is one of those $\tau$s, that is, the name of a potential member of the thing in $M[G]$ that $\sigma$ names.
This $\pi$ too is a name, so $\pi\in\sigma$ is a well-formed proposition in the forcing language.
It is possible for $\pi\in\operatorname{dom}(\sigma)$ and yet $M[G]\not\models \pi\in\sigma$, namely if $\langle \pi,q\rangle\in\sigma$ for some $q$ that is not in $G$.
Conversely, $\pi\in\sigma$ can be true in $M[G]$ even though $\pi\notin\operatorname{dom}(\sigma)$, namely if $\pi$ happens to name the same individual in $M[G]$ that some other name in $\operatorname{dom}(\sigma)$ also names.
