It’s smarter perhaps to pack a bed that’s inflatable (3,8)

I believe the answer is:

air mattress

I'm a little stuck... Click here to teach me more about this clue!

'a bed that's inflatable' is the definition.
I can't judge whether this defines the answer.

'it's smarter perhaps to pack' is the wordplay.
'perhaps' is an anagram indicator.
'to' says to put letters next to each other (I've seen this in other clues).
'pack' becomes 'ai' (I am not sure about this - if you are sure you should believe this answer much more).
'ts'+'smarter'='tssmarter'
'tssmarter' with letters rearranged gives 'rmattress'.
'rmattress' after 'ai' is 'AIR MATTRESS'.

Can you help me to learn more?

I've seen this clue in The Hindu.

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