We can use a full infinitive or a gerund as a complement of the 'be'-verb. For example,
The first thing I do in the morning is to check my mobile phone.
The first thing I do in the morning is checking my mobile phone.
But I've seen the bare infinitive used in such construction with no real difference:
The first thing I do in the morning is check my mobile phone.
Thus, bare infinitive is used when the finite element of the first verb is a form of "do" : Using bare infinitive after 'does'.
Now, my question is whether it's a case of elision, such as "The first thing I do in the morning is [that I] check my mobile phone." or is there any other grammatical explanation?