Consequences of UP equals NP
It is known that $\mathsf{UP= NP}$ implies $\mathsf{SpanP = \#P}$ since Kobler, Schoning, and Toran proved that $\mathsf{UP= NP}$ if and only if $\mathsf{SpanP = \#P}$. It is easy to see that $\mathsf {\#P}$ is contained in $\mathsf {SpanP}$.

A function $f : Σ^* →\mathbb N$ is in $\mathsf{SpanP}$ if there is an $\mathsf {NP}$ Turing machine transducer $M$ such that for all $x$, $f(x)$ is the number of distinct outputs of $M$ on input $x$.

J. Kobler, U. Schoning, and J. Toran. On counting and approximation, Acta Informatica,
26:363-379, 1989.