"The four houses are called Gryffindor, Hufflepuff, Ravenclaw, and Slytherin. Each house has its own noble history and each has produced outstanding witches and wizards. While you are at Hogwarts, your triumphs will earn your house points, while any rule-breaking will lose house points. At the end of the year, the house with the most points is awarded the house cup, a great honor. I hope each of you will be a credit to whichever house becomes yours.
(Harry Potter and the Sorcerer’s Stone, pp.114)
When I read the phrase, I feel like hopping onto the next stepping stone, in between a stone is missed. So I account if the phrase were ‘a credit to whichever house that becomes yours’, I wouldn’t have felt that way.
In this phrase, is there missing subject as I mentioned, or does the phrase itself make the complement of ‘to’? I mean is it the way how English users say?