Description:
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The properties of the Mysqli object already contain server-info, server-version, errors, etc., but not the current (selected) database.
It would be great if dbname could be added to mysqli properties.
With the property it gets easier to check if the right database was selected or if none was selected.

Test script:
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PHP code looks like this:

class mysqli_db extends mysqli {
  public function __construct($host=null, $user=null, $pw=null, $db=null) {
    call_user_func_array("parent::__construct", func_get_args());
    $this->db = $db;
  }
  // same for real_connect() ...

  public function select_db($db) {
    parent::select_db($db);
    $this->db = $db;
  }
}

$db = new mysqli_db('127.0.0.1', 'root', '', 'test');
echo $db->db; // test

$db->select_db('test2');
echo $db->db; // test2