As I have argued below and elsewhere, the copula 'is', as in
(1) Tom is running
has a twofold function. The first is to join subject and predicate together into a single content that is expressible by the sentence. For example 'Tom's running' or 'that Tom runs'. The second - and this is something only a verb can do - is actually to express this content. Thus
(2) Tom's running is the case
or 'That Tom runs is a fact' or 'it is true that Tom runs'. Now as it happens the operator 'it is true' which we attach to the that-clause 'that Tom runs' or 'is the case' which we attach to 'Tom's running' also contains the verb 'is'. But this, I argue, is a grammatical accident. The 'is' in 'is the case' is mere filling. The operator 'is the case' is a unitary element of the sentence. It would be better expressed by a single verb 'isthecase' or 'itistrue'. If it is essentially complex, then of course it contains the verb 'is', and if it functions as a copula, then it will have the twofold function described above. 'the case' would be a predicate that we could attach to the subject 'Tom's running' to form the noun-phrase 'Tom's running being the case'. And we could further assert this to form a new sentence 'Tom's running being the case is the case' and so on ad infinitum.
This is obvious with 'is the case'. We are not tempted into supposing that 'the case' is a predicate that qualifies Tom's running in any way. It is less obvious with 'it is true'. It is more tempting to suppose that the noun-phrase 'that Tom is running' refers to some content, a thing, and that in forming 'that Tom is running is true' we are predicating 'truth' of this content. Bad things lie in that direction.
See also Maverick Philosopher for a different take on why truth is not a predicate