Image of Inverse Image
Jump to navigation
Jump to search
Theorem
Let $S, T$ be sets.
Let $f: S \to T$ be a mapping.
Let $X$ be a subset of $T$.
Then:
- $f \sqbrk {f^{-1} \sqbrk X} \subseteq X$
where:
Proof
Let $x \in f \sqbrk {f^{-1} \sqbrk X}$.
By definition of image of set:
- $\exists y \in S: y \in f^{-1} \sqbrk X \land x = \map f y$
By definition of image of set under relation:
- $\map f y \in X$
Thus $x \in X$
$\blacksquare$
Sources
- Mizar article FUNCT_1:75