Vain cat destroys Pope’s city (7)

I believe the answer is:

vatican

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

'pope's city' is the definition.
'vatican' can be an answer for 'city' (I've seen this before). I'm unsure of the 'pope's' bit.

'vain cat destroys' is the wordplay.
'destroys' is an anagram indicator (I've seen 'destroyed' mean this).
'vain'+'cat'='vaincat'
'vaincat' anagrammed gives 'VATICAN'.

Can you help me to learn more?

(Other definitions for vatican that I've seen before include "Palace of the popes in Rome" , "religious authority" , "Holy See" , "Papacy" , "ecclesiastical centre" .)

I've seen this clue in The Mirror.

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