Functional tests depth of assertions

I'm developing a project based on Symfony2 and PHPUnit. As far as I know functional tests represents what the system should do in a user perspective. But, I have some questions about this approach.

Suppose I'm testing an user registration form. So the first thing that I should to do after submit the form is to assert that the response was successful, an email was sent and probably make assertions on success page. Internally the system should store the registration date, change the user status and etc.

The question is: checking this low level of code like registration date and status should be covered by functional tests? If not, where is the best place to put this kind of tests?