Reference for theorem? Inequality of integrals of increasing function over two distributions

Question

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$.

This seems like it must be true: $$ \int_0^{\infty} H(x) dF_1 \geq \int_0^{\infty} H(x) dF_2 $$

An intuition is: everywhere that $F_2$ has more mass/density than $F_1$ can be mapped "rightward" to a place where $F_1$ has this same amount more than $F_2$; and the contribution to the integral will be higher in the latter case because $H$ is increasing.

But this seems a bit annoying to formalize; is there an easy proof or name for the theorem (or reference/textbook)? Thanks!

Answer 1

As Nate Eldredge said in the comment, this condition is called stochastic dominance and my claim is true.