You don't have to "prove" anything. That's like asking a person in a wheelchair to "prove" they cannot walk. The establishment is not supposed to even ask what your "disability is" and there are harsh consequences for violating the law.
Yorkies and small dogs are used a lot as service animals because they are so small and they travel well and people with physical disabilities like arthritis can carry them comfortably. Pick up the latest issue of Yorkshire Terrier Magazine nd there is a long article in there about yorkies who are service animals, some who have saved lives. There is a link to a Question and Answer paper for business establishment that I carry with me to educate businesses that are unaware of the law.
Here is the link for the business questions and answers regarding Service Animals:
http://www.usdoj.gov/crt/ada/qasrvc.htm