Mine’s: A possessive of a possessive

Your dog's collar is blue.
Mine's is red.

Is that grammatical?

Or did I just invent a new grammatical category: the possessive form of a possessive pronoun?

Is there some connection with

a friend of John's

I think it’s more likely to occur than

a friend of John

But it would be

the collar of my dog

rather than

the collar of my dog's

See HairOfTheDog

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