Correlation between sanity testing and regression testing

After reading about many posts about smoke and sanity testing i have concluded following about testing order:

Smoke testing(generalized health checkup)---then--> Sanity Testing(checking some main functionalities to somewhat more deeper level)(Specialized health checkup)----------------then--------> Functional testing(full functionality checking at deeper levels)

In many posts i have read that sanity is subset of regression testing. But according to above mentioned order it seems that sanity is subset of functional testing. Can anyone elaborate that how sanity is subset of regression testing?