Auxiliary Relation Image of Element is Upper Section
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Theorem
Let $L = \struct {S, \preceq}$ be an ordered set.
Let $R$ be an auxiliary relation on $S$.
Let $x \in S$.
Then $\map R x$ is an upper section
where $\map R x$ denotes the image of $x$ under $R$.
Proof
Let $a \in \map R x, b \in S$ such that
- $a \preceq b$
By definition of $R$-image of element:
- $\tuple {x, a} \in R$
By definition of reflexivity:
- $x \preceq x$
By definition of auxiliary relation:
- $\tuple {x, b} \in R$
Thus by definition of $R$-image of element:
- $b \in \map R x$
$\blacksquare$
Sources
- Mizar article WAYBEL_7:funcreg 3