Theorem efmndbasfi

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 )

Proof

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 )