Quiet person chops Cheddar cheese (9)

I believe the answer is:

mousetrap

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'cheese' is the definition.
(I've seen this before)

'quiet person chops cheddar' is the wordplay.
'quiet person' becomes 'mouse' (I've seen this before).
'chops cheddar' becomes 'trap' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'mouse'+'trap'='MOUSETRAP'

Can you help me to learn more?

(Other definitions for mousetrap that I've seen before include "For instance, old Cheshire" , "Snare for vermin" , "Cheap cheese" , "pest control device" , "Anti-rodent device" .)

I've seen this clue in The Independent.

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