In a paper I encountered the following notation
$$P(Z\leq z,u\leq Y\leq v)=C(F_{Z}(z),F_{Y}(v)-F_{Y}(u))$$
However I don't see why this holds in relation to uniform random variables. Usually $$P(Z\leq,Y\leq v)=P(F_{Z}(Z)\leq F_{Z}(z),F_{Y}(Y)\leq F_{Y}(v))=P(U_{1}\leq F_{Z}(z), U_{2}\leq F_{Y}(v))=C(F_{Z}(z),F_{Y}(v))$$
But probability above i would write $$P(Z\leq z,u\leq Y\leq v)=P(F_{Z}(Z)\leq F_{Z}(z),F_{Y}(Y)\leq F_{Y}(v))-P(F_{Z}(Z)\leq F_{Z}(z),F_{Y}(Y)\leq F_{Y}(u))$$ and then use copulas.
Can anyone explain to me where the copula $C(F_{Z}(z),F_{Y}(v)-F_{Y}(u))$ comes from in terms of uniform random variables?