Most definitions of a k$k$-wise independent family of hash functions I have encountered state that a family $H$ of hash functions from $D$ to $R$ is k-wise independent if for all distinct $x_1, x_2... x_k \in D$$x_1, x_2,\dots, x_k \in D$ and $y_1, y_2 ... y_k \in R$$y_1, y_2,\dots, y_k \in R$,
$$\mathbb{P}_{h \in H}(h(x_1) = y_1, h(x_2) = y_2, ..., h(x_k) = y_k) = \frac{1}{|R|^k}$$$$\mathbb{P}_{h \in H}(h(x_1) = y_1, h(x_2) = y_2, \dots, h(x_k) = y_k) = \frac{1}{|R|^k}$$
The Wikipedia article on k-wise independent hash functions (which uses the above definition) claims that the definition is equivalent to the following two conditions:
(i) For all $x \in D$, $h(x)$ is uniformly distributed in $R$ given that $h$ is randomly chosen from $H$.
(ii) For any fixed distinct keys $x_1, x_2... x_k \in D$$x_1, x_2,\dots, x_k \in D$, as h$h$ is randomly drawn from $H$, the hash codes $h(x_1), h(x_2), ..., h(x_k)$$h(x_1), h(x_2), \dots, h(x_k)$ are independent random variables.
It is not obvious to me how one proves (i) from the above definition without explicitly assuming (ii) in the definition as well (and vice-versa). How is the definition sufficient for proving both (i) and (ii)?
