JMO 1992 Problem 4

Let $A$ be a $m\times n\ (m\neq n)$ matrix with the entries $0$ and $1$. Suppose that if $f$ is injective such that $f: \{1,\ 2,\ \cdots ,\ m\}\longrightarrow \{1,\ 2,\ \cdots ,\ n\}$, then there exists $1\leq i\leq m$ such that $(i,\ f(i))$ element is $0$.

Prove that there exist $S\subseteq \{1,\ 2,\ \cdots ,\ m\}$ and $T\subseteq \{1,\ 2,\ \cdots ,\ n\}$ satisfying the condition:

$i)$ if $i\in{S},\ j\in{T}$, then $(i,\ j)$ entry is $0$.

$ii)$ $card\ S + card\ T > n$.