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

'self-importantly' is the definition.
I know nothing about this answer so I cannot 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 found within the answer.
'and' could be 'n' (common abbreviation for 'and') and 'n' is located in the answer.
'very' could be 'v' (common 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?