A question about of $\mathbb{P}$-name Let $ \mathbb{P}$ be poset and $\dot{Q}$ a $\mathbb{P}$-name. 
Question: we can find some  cardinal $\mu$  in  the ground model such that $ \Vdash |\dot{Q}| \leq |\mu|$.?

Of course. Take $\mu=|\dot Q|$.
Define in the ground model $\dot Q$ as $(p_\alpha,\dot q_\alpha)$, and define the map $\dot f=\{(1_\Bbb P,(\check\alpha,\dot q_\alpha)^\bullet)\mid\alpha<\mu\}$ and show that $\Vdash\dot Q\subseteq\operatorname{rng}(\dot f)$.
