Intersection of dense subset with open subset is dense in the open subset

Statement
Suppose $$X$$ is a topological space, $$U$$ is an open subset of $$X$$, and $$D$$ is a dense subset of $$X$$. Then, $$U \cap D$$ is a dense subset of $$U$$ equipped with the subspace topology.

Facts used

 * 1) uses::Open subset of open subspace is open

Proof
Given: $$X$$ is a topological space, $$U$$ is an open subset of $$X$$, and $$D$$ is a dense subset of $$X$$. $$V$$ is a non-empty open subset of $$U$$ (in the subspace topology).

To prove: $$V \cap (U \cap D)$$ is non-empty.

Proof: