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. 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:

Step no.	Assertion	Facts used	Given data used	Previous steps used	Explanation
1	$V$ is open in $X$	Fact (1)	$V$ open in $U$, $U$ open in $X$
2	$V\cap D$ is non-empty		$V$ is non-empty, $D$ is dense	Step (1)	Step-given direct
3	$V\cap D=V\cap (U\cap D)$		$V\subseteq U$
4	$V\cap (U\cap D)$ is non-empty			Steps (2), (3)	Step-combination direct