Suppose that \kappa is a regular cardinal such that 2^{\lambda} < \kappa for all \lambda < \kappa. Prove that H(\kappa)=\mathcal{V}_{\kappa}. Which axioms of ZFC does \mathcal{V}_{\kappa} satisfy?

Here, for a given cardinal \kappa, H(\kappa) denotes the collection of sets whose transitive closure has cardinality less than \kappa. Also, a well orderable infinite cardinal \kappa is regular if \text{cf}(\kappa) =_c \kappa, where \text{cf} denotes the cofinality. The set \mathcal{V}_{\alpha} is defined by \mathcal{V}_{0}= \emptyset, \mathcal{V}_{\alpha+1}=\mathcal{P}(\mathcal{V}_{\a  lpha}), and \mathcal{V}_{\lambda}= \bigcup_{\alpha< \lambda} \mathcal{V}_{\alpha} if \text{Limit}(\lambda).

I am not sure how to prove this. I would appreciate a few hints or suggestions. Thank you.

Zermelo?Fraenkel set theory - Wikipedia, the free encyclopedia