I'm currently reading Keisler's Elementary Calculus -- An Infinitesimal Approach, which develops the main results usually thought in undergrad calculus using Robinson's hyperreal numbers (instead of the more common "$\epsilon - \delta$ approach"). After the demonstration of the Fundamental Theorem of Calculus---which shows that the function $F(x) = \int_a^x f(t)dt$ is an antiderivative of the function $f$---it is just stated that the family of antiderivatives of the function $f$ will be denoted as $\int f(x)dx$. My question is, why does this notation work? In particular when doing integration by substitution, it is clear that the notation works "as expected" (lacking a better way to describe it).
To put the question another way, supposing I had just discovered the Fundamental Theorem of Calculus, how should I then reason in order to conclude that the notation $\int f(x)dx$ is a good one?