Cardinality of Image of Set not greater than Cardinality of Set
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Theorem
Let $X, Y$ be sets.
Let $f: X \to Y$ be a mapping.
Let $A$ be a subset of $X$.
Then:
- $\card {\map {f^\to} A} \le \card A$
where $\card A$ denotes the cardinality of $A$.
Proof
By definitions of surjection and restriction of mapping:
- $F \restriction_A: A \to \map {f^\to} A$ is a surjection
Thus by Surjection iff Cardinal Inequality:
- $\card {\map {f^\to} A} \le \card A$
$\blacksquare$
Sources
- Mizar article CARD_1:67