Expectation of the minimum of two continuous random variables - using the joint pdf

Define $$Z = \min(X, Y)$$ and the joint pdf of $$X$$ and $$Y$$ as $$f_{XY}(x,y)$$.

I saw an approach that said

$$E[Z] = \int \int \min(x,y) f_{XY}(x,y) \,dy\,dx$$

Is this readily obvious, or do you need to convert the following: $$E[Z] = \int \min(x,y)f_Z(z) \,dz$$

to the above?