Waving a fan and behaving very self-importantly (6,7,4)

'self-importantly' is the definition.
I don't know anything about this answer so I can't tell whether this works.

'waving a fan and behaving very' is the wordplay.
I cannot really understand how this works, but
'a' could be 'one' and 'one' is present in the answer.
'and' could be 'n' (common abbreviation for 'and') and 'n' is found in the answer.
'very' could be 'v' (abbreviation) and 'v' is present in the answer.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?