TL;DR: Your assumption is correct, "the new relation" is the main subject, while "result of the expression" is the nominal predicate.
It's a remnant of the ancient Essive/Translative grammatical cases that existed in older languages, retained in some modern languages (Uralic family), but is obsolete (converted to Instrumental) in modern Slavonic languages.
What's going on here?
First things first. The entire sentence translates to
The result of the expression is the new relation
For sake of simplicity, let me simplify it to:
{The} result is {the} relation
Thinking in a paradigm of Romance languages, one may think that is here denotes equivalence, hence "result is relation" and "relation is result" are equivalent statements. This happens because the verb to be in English has several essentially different meanings: be_existence, be_location, be_identity/attribute, be_possession, be_to_mean.
Of the listed meanings, we see that only identity/attribute satisfies our need, and the relation is not symmetric: in "light turns red", only light can be the direct subject.
Note: a Nominative-Instrumental switch for nonverbial predicates occurs only with dynamic be_identity verb. Consider this:
- ona jest nauczycielka — she is a teacher — stative predication (if you really meant to imply that a profession is an integral part of a person) → Nominative;
- ona jest nauczycielką — she {currently} is a teacher — dynamic predication (over a certain period of time) → Instrumental;
- she wants to become a teacher — dynamic predication (transition) → Instrumental;
- she was a teacher — dynamic predication (because it is associated with a certain period of time in the past) → Instrumental;
- she will be a teacher — dynamic predication (same as above) → Instrumental;
Returning back to the original sentence, "the new relation" is the subject, and "the result of the expression" (a compound nominal predicate) acts as a state from which "the new relation" arises from.
Note: the word order is irrelevant here. IMO, it's just a matter of stylistic: you put the cause first and the effect after.
But why Instrumental, finally?
In all examples above, the subject is (or temporary was, or becomes) in a certain state. Ancient languages had Essive and Translative cases to denote this relation.
Finnish and Estonian have retained it over the course of time, while others, including Polish, have lost these cases, and began using the Instrumental case for encoding the dynamic relation of the Subject and its Nominal/Adjectival predicates.
P.S. I think I should add some references to academic works about the Nominative-Instrumental switch in Polish, but I can't recall any off the top of my mind (TODO).