Let $\mathbb{P}$ be a poset and $B,D$ be sets.
Let $p \in \mathbb{P}$ and $\sigma$ be a $\mathbb{P}$-name such that $p \Vdash \sigma \in B$. Then there exist a nice name $\tau$ for an object in $B$ such that $p \Vdash \sigma = \tau.$
Also if $\mu$ be a $\mathbb{P}$-name such that $p \Vdash \mu:D\to B$ Then there exist a nice name $\pi$ for a function from $D$ into $B$ such that $p \Vdash \mu = \pi.$
I am studying the book Kunen and I'm a little confused when defining a $\mathbb{P}$-name.
A suggestion of how to define a nice name. Thanks
A nice name can be defined as followed:
Definition IV.3.8 (In the 2013 edition of the Kunen) For $\tau \in V^{\mathbb{P}}$, a nice name for a subset of $\tau$ is a name of the form $\cup \{ \{\sigma \} \times A_{\sigma} | \sigma \in dom(\tau)\}$, where each $A_\sigma$ is an antichain in $\mathbb{P}$.
So a nice name for a subset of $\tau$ is one where you look at $\mathbb{P}$-names in the domain of $\tau$ (i.e. candidates for members of subsets of $\tau$), index antichains in $\mathbb{P}$ using these names, and then look at the set of ordered pairs you can make whose first coordinates are each $\sigma$, and second coordinates are the members of the antichain indeced by $\sigma$.
Essentially, you want to have names that interact nicely with the antichains in $\mathbb{P}$. The lemmas you mention show that you can find nice names that will do the job of normal $\mathbb{P}$-names, but give you a little more traction on what's going on with the antichains in $\mathbb{P}$ during a forcing construction. Remember any $\mathbb{P}$-generic over $\mathfrak{M}$ has to intersect a maximal antichain of $\mathbb{P}$. So, by using a nice name $\theta$ for a subset of $\tau$ (and where the antichain $A_{\sigma}$ is maximal, you're guaranteeing that for each $\sigma \in dom(\theta)$, $\sigma_G$ makes it into $\theta_G \subseteq \tau_G$, as the generic can't avoid including it.