# Counting number of functions

I am reading the paper here and have a small doubt in Lemma 1. The proof (on page 1) begins with:

Lemma 1. Let $M$ and $N$ be cardinal numbers. Let $S$ be $N$-regular in $X$, a set of cardinality $M$, and let $T$ be $N^M$-regular in $Y$. Then if $S\otimes T$ is N-regular in $X\times Y$.

Proof: Let $P$ be a set of cardinality $N$. Then a partition of $X \times Y$ into $N$ parts can be represented by a function from $X \times Y$ into $P$. For each $y \in Y$, $f$ defines a function $f_y$ from $X$ into $P$ given by $f_y(x) = f(x,y)$. Since there are $N^M$ such functions the mapping $y \rightarrow f_y$ induces a partition of $Y$ into $N^M$ parts.

My doubt is that the proof should not the proof say "Since there are at least $N^M$ such functions the mapping $y \rightarrow f_y$ induces a partition of $Y$ into at least $N^M$ parts." since each of the $N^M$ functions from $X$ to $P$ may correspond to more then 1 $f_y$.

If my doubt is correct (and I expect that the proof will still go through) then should small omissions like this not be entirely unexpected in research papers. I am asking this because I have little experience in reading research papers.

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The point is that the entire set of functions from $X$ to $P$ has cardinality $N^M$, since $|X|=M$ and $|P|=N$. Let $\mathscr{G}$ be the set of these functions. For each $g\in\mathscr{G}$ let $Y_g=\{y\in Y:f_y=g\}$. Then $\mathscr{P}=\{Y_g:g\in\mathscr{G}\}$ is a partition of $Y$ into $|\mathscr{G}|=N^M$ parts. (Actually, some of these parts may be empty, so if your definition of partition doesn’t allow you to count empty parts, you should say that $\mathscr{P}$ is a partition of $Y$ into at most $N^M$ parts.)
It’s perfectly true that there may be distinct $y_0,y_1\in Y$ such that $f_{y_0}=f_{y_1}$, but that just says that $f_{y_0}$ and $f_{y_1}$ belong to the same part of $\mathscr{P}$. There is no possibility that $\mathscr{P}$ has more than $N^M$ parts, since $\mathscr{G}$ has only $N^M$ members.