Image of Subset under Relation is Subset of Image/Corollary 3

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Corollary of Image of Subset under Relation is Subset of Image

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping from $S$ to $T$.


Let $C, D \subseteq T$.


Then:

$C \subseteq D \implies f^{-1} \sqbrk C \subseteq f^{-1} \sqbrk D$


This can be expressed in the language and notation of inverse image mappings as:

$\forall C, D \in \powerset T: C \subseteq D \implies \map {f^\gets} C \subseteq \map {f^\gets} D$


Proof

As $f: S \to T$ is a mapping, it is also a relation, and thus so is its inverse:

$f^{-1} \subseteq T \times S$

The result follows directly from Image of Subset under Relation is Subset of Image.

$\blacksquare$


Sources