Day church converted Austrian prince’s estate (9)

I believe the answer is:

archduchy

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'austrian prince's estate' is the definition.
Although both the answer and definition are singular nouns, I can't see how they can define each other.

'day church converted' is the wordplay.
'converted' is an anagram indicator.
'day'+'church'='daychurch'
'daychurch' with letters rearranged gives 'ARCHDUCHY'.

Can you help me to learn more?

(Another definition for archduchy that I've seen is " Austria, say".)

I've seen this clue in The Times.

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