Description: The monoid of endofunctions on a finite set A is finite. (Contributed by AV, 27-Jan-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | efmndbas.g | |- G = ( EndoFMnd ` A ) |
|
efmndbas.b | |- B = ( Base ` G ) |
||
Assertion | efmndbasfi | |- ( A e. Fin -> B e. Fin ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efmndbas.g | |- G = ( EndoFMnd ` A ) |
|
2 | efmndbas.b | |- B = ( Base ` G ) |
|
3 | 1 2 | efmndbas | |- B = ( A ^m A ) |
4 | mapfi | |- ( ( A e. Fin /\ A e. Fin ) -> ( A ^m A ) e. Fin ) |
|
5 | 4 | anidms | |- ( A e. Fin -> ( A ^m A ) e. Fin ) |
6 | 3 5 | eqeltrid | |- ( A e. Fin -> B e. Fin ) |