The rules for "a" and "the" can be complex and subtle. Yes, in general, "the X" is used when there is one specific X that is being talked about, and the speaker and the audience know which one it is. And yes, often the reason they know is because it was already mentioned (and may originally have been "a/an X" instead).
But that is not what is happening here.
CowperKettle's answer is basically correct. However, it isn't quite right in how it describes the "she's the X" idiom.
A hypothetical conversation:
Alice: "You're an idiot."
Bob: "You're the idiot!"
Why has Bob switched from "an idiot" to "the idiot"? The reason is all about what's implied, not what is said explicitly.
Bob is implying that there is only one idiot in this conversation. He has not explicitly denied being an idiot, but by saying "the idiot", he implies that there is only one (and so he's not it).
(In practice, this might be happening in a situation where Bob has done something quite idiotic, so it would be unreasonable for him to completely deny being an idiot. Bob's implication, then, is that compared to him, Alice is much more of an idiot.)
Summary: In this case, switching from "a" to "the" is not done to follow any particular rules of English. It is a deliberate choice of phrasing that puts a lot of implied meaning into a very short sentence. The effect is to deny being a liar, or suggest that "she" is a much worse liar than the narrator.
Also note that, in speech, the emphasis shouldn't be on the article. If "she's the liar" was spoken aloud, the emphasis would be on "she's", not on "the".
I mention this because when I read CowperKettle's answer, which bolds "a" and "the", I read it as having emphasis on those words, which sounds strange to me. (I would guess that CowperKettle bolded those words because they're the ones we're talking about, not because they should be emphasised in speech. And that's fine. But another common use of bold is to mark emphasis, and so I thought this warning was worth including.)