Image of Mapping from Finite Set is Finite
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Theorem
Let $X, Y$ be sets.
Let $f: X \to Y$ be a mapping.
Let $X$ be a finite set.
Then $f \sqbrk X$ is a finite set.
Proof
By definition of surjection:
- $f: X \to f \sqbrk X$ is a surjection.
The case when $X \ne \O$:
By Surjection iff Cardinal Inequality:
- $\card {f \sqbrk X} \le \card X$
Thus by Set of Cardinality not Greater than Cardinality of Finite Set is Finite:
- $f \sqbrk X$ is finite.
$\Box$
The case when $X = \O$:
By a corollary to Image of Empty Set is Empty Set:
- $f \sqbrk X = \O$
Thus the result holds.
$\blacksquare$
Sources
- Mizar article FINSET_1:8