Articles and pronouns

So as I’ve begun my study of German I’ve noticed that the definite article and indefinite article have some pronoun-like uses. I know that the definite article can be used as a relative pronoun, but there are some other things I’ve seen that I haven’t been able to understand, like when someone says about someone, “Der is tot.” Why der, and not er? Are there are any other uses like this?

You can say “Er ist tot.” (He is dead.). It doesn’t change much.
If you use “der, die, das” you are omitting a noun. Der, Die, Das … ist tot. (The … is dead. / That one is dead.)

For example:
Der Hund ist tot. → Der ist tot. (The dog is dead. / That one is dead.) OR Er ist tot. (He is dead.)
Die Katze ist tot. → Die ist tot. (The cat is dead. / That one is dead.) OR Sie ist tot. (She is dead.)
Das Tier ist tot. → Das ist tot. (The animal is dead. / That one is dead.) OR Es ist tot. (It is dead.)

You can omit the noun, if it is known from context, obvious, like pointing at something, and so on.
Depending on the context one version might sound slightly more natural than the other, but the meaning is more or less the same. Nothing to worry about.

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Thank you.