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Let $F$ is an $n$-ary relation (with $n$ being any index set).

Can the following formula be simplified? $$(\lambda x\in n:s(x))\in F$$ ($s$ is some function).

Here $\lambda$ is defined as: $$(\lambda x\in D:f(x)) = \{(x;f(x)) | x\in D\}$$ for every set $D$ and formula $f(x)$ dependent on a variable $x$.

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up vote 5 down vote accepted

I prefer to reserve the letter $n$ for integers, so I’ll use $I$ for the index set.

If $F$ is an $I$-ary relation on $X$, where $I$ is an arbitrary index set, then $F$ is simply a subset of $X^I$, which can be viewed as the set of functions from $I$ to $X$.

The function $s$ is the set $\{\langle i,s(i)\rangle:i\in I\}$, so all you’re saying is that $s\in F$.

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