Joint distribution of random variable with an order statistic

It is obvious that for every $j\in\{1,\dots,n\}$:$$P\left(X_{\left(1\right)}\leq u,X_{j}\leq v\right)=P\left(X_{\left(1\right)}\leq u,X_{1}\leq v\right)$$

If $v\leq u$ then $\left\{ X_{\left(1\right)}\leq u,X_{1}\leq v\right\} =\left\{ X_{1}\leq v\right\} $ so that in that case: $$P\left(X_{\left(1\right)}\leq u,X_{1}\leq v\right)=P\left(X_{1}\leq v\right)=F_{X}\left(v\right)$$

If $v>u$ then we can go for:

$$\begin{aligned}P\left(X_{\left(1\right)}\leq u,X_{1}\leq v\right) & =P\left(X_{\left(1\right)}\leq u,X_{1}\leq u\right)+P\left(X_{\left(1\right)}\leq u,u<X_{1}\leq v\right)\\ & =P\left(X_{1}\leq u\right)+P\left(\min\left(X_{2},\dots,X_{n}\right)\leq u,u<X_{1}\leq v\right)\\ & =P\left(X_{1}\leq u\right)+P\left(\min\left(X_{2},\dots,X_{n}\right)\leq u\right)P\left(u<X_{1}\leq v\right)\\ & =F_{X}\left(u\right)+\left(1-\left(1-F_{X}\left(u\right)\right)^{n-1}\right)\left(F_{X}\left(v\right)-F_{X}\left(u\right)\right) \end{aligned} $$

From @drhab's answer we see that the joint distribution of $(X_{(1)},X_j)$ is given by $$ G(u,v) = F(v)\mathsf 1_{\{v\leqslant u\}} + \left(1-(1-F(u))^{n-1}\right)(F(v)-F(u))\mathsf 1_{\{v>u\}}. $$ For a concrete example, let $X_n\stackrel{\mathrm{i.i.d.}}{\sim}\mathrm{Expo}(\lambda)$, that is, $F(t) = \left(1 - e^{-\lambda t}\right)\mathsf 1_{(0,\infty)}(t)$. Then the joint distribution of $(X_{(1)},X_j)$ is given by \begin{align} G(u,v) =& F(v)\mathsf 1_{\{v\leqslant u\}} + \left(1-(1-F(u))^{n-1}\right)(F(v)-F(u))\mathsf 1_{\{v>u\}}\\ &=\begin{cases} 1-e^{-\lambda v},& v\leqslant u\\ \left(1-e^{-(n-1)\lambda u}\right)\left(e^{-\lambda u}-e^{-\lambda v}\right),& v>0. \end{cases} \end{align}