# Definition talk:Image (Set Theory)/Mapping/Element

Without the union sign, it is the collection of all valid second elements--namely, $\left\{ t \right\}$. It is the collection of all $t$ such that $( s,t ) \in \mathcal R$, so the resulting set has $t$ as its only member. I'm just using the union sign to make it functionally correct, even if the definition is not intuitive.
To make the definition more intuitive, you could introduce (as per Whitehead/Russell or Quine) the notation $( \iota x P(x) )$ to mean "the sole $x$ such that $P(x)$". It could default to $\varnothing$ if $P(x)$ had no unique $x$ satisfying it. $f(s)$ would become $( \iota t ( s,t ) \in f )$. --Andrew Salmon 18:45, 10 August 2012 (UTC)