Let $\mathbb{P}$ be poset.
Let $D, B$ be sets. We say that a $\mathbb{P}$-name $\dot{z}$ is a nice name for a function from $D$ into $B$ if there is $\left<{A_{d},h_{d}}\right>_{d \in D}$ such that $\dot{z}(d)=\left<{A_{d},h_{d}}\right>$(as a nice name) for all $d\in D$, that is, $A_{d}$ is a maximal antichain and $h_{d}:A_{d}\to B$ such that $ p\Vdash\dot{z}(d)=h_{d}(p)$ for all $p \in A_{d}$.
Let $B$ be a set. We say that a $\mathbb{P}$-name $\dot{b}$ is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function $h:A\rightarrow{B}$ such that $ p\Vdash\dot{b}=h(p)$ for all $p\in{A}$. Here, we abuse of the notation and say $\dot{b}=\left<{A,h}\right>$ .
Given a maximal antichain $A$ and a function $h:A\rightarrow{B}$, we can construct a name $\dot{b}$ such that $\dot{b}=\left<{A,h}\right>$ applying the mixing lemma:
Let $A$ be an antichain in $\mathbb{P}$ and suppose that $\tau_p$ for $p\in A$ are $\mathbb{P}$-names. Then there is a $\mathbb{P}$-name $\sigma$ such that $p\Vdash \sigma=\tau_p$ for each $p\in A$.
For $d \in D$, if $A_{d}$ is a maximal antichain and $h_{d}:A_{d}\to B$ is possible to construct a $\dot{z}$ such that $\dot{z}=\left<{A_{d},h_{d}}\right>$ applying the mixing lemma.?
If not how I can build such $\dot{z}$ such that $\dot{z}=\left<{A_{d},h_{d}}\right>$. Any ideas thanks
Then use $(a)$ to build $\dot{z}=\left<{A_{d},h_{d}}\right>$