Equivalent Definition of p-Sweep Outs

Question

Let $M$ be a compact Riemannian manifold of dimension $n+1$ and let $\mathcal{Z}_{n}(M;\mathbb{Z}_2)$ denote the space of codim one flat cycles in $M$. This space is weakly homotopic to $\mathbb{R}P^{\infty}$ and we have $H^1(\mathcal{Z}_{n}(M;\mathbb{Z}_2);\mathbb{Z}_2)=\mathbb{Z}_2$. Let $\lambda$ denote the generator of $H^1(\mathcal{Z}_{n}(M;\mathbb{Z}_2);\mathbb{Z}_2)$. We say a map $\Phi\colon X\to\mathcal{Z}_{n}(M;\mathbb{Z}_2)$, where $X$ is a simplicial complex, is a $p$-sweep out if $\Phi^*(\lambda^p)$ is not zero. I am trying to prove being a $p$-sweep out is equivalent to the fact that for every $p$-tuples of points $(m_1,m_2,\dots,m_p)$ in $M$, there exists $x$ in $X$ such that $m_i$ is in support of $\Phi(x)$ for every $i$. Do you know any proof or reference for this?