Subset of Codomain is Superset of Image of Preimage/Proof 2
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Theorem
Let $f: S \to T$ be a mapping.
Then:
- $B \subseteq T \implies \paren {f \circ f^{-1} } \sqbrk B \subseteq B$
where:
- $f \sqbrk B$ denotes the image of $B$ under $f$
- $f^{-1}$ denotes the inverse of $f$
- $f \circ f^{-1}$ denotes composition of $f$ and $f^{-1}$.
This can be expressed in the language and notation of direct image mappings and inverse image mappings as:
- $\forall B \in \powerset T: \map {\paren {f^\to \circ f^\gets} } B \subseteq B$
Proof
Let $y \in B$.
Then:
- $\exists x \in S: y = f \left({x}\right)$
Therefore by definition of preimage of subset:
- $\exists x \in f^{-1} \left[{B}\right]$
It follows by definition of image of subset that:
- $y \in f \left[{f^{-1} \left[{B}\right]}\right]$
Thus by definition of composition $f$ with $f^{-1}$:
- $y \in \left({f \circ f^{-1}}\right) \left[{B}\right]$
The result follows by definition of subset.
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites