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	$$⊢G=\mathrm{EndoFMnd}\left(A\right)$$
	efmndbas.b	$$⊢B={\mathrm{Base}}_G$$
Assertion	efmndbasfi	$$⊢A\in \mathrm{Fin}\to B\in \mathrm{Fin}$$

Proof

Step	Hyp	Ref	Expression
1		efmndbas.g	$$⊢G=\mathrm{EndoFMnd}\left(A\right)$$
2		efmndbas.b	$$⊢B={\mathrm{Base}}_G$$
3	1 2	efmndbas	$$⊢B=A^A$$
4		mapfi	$$⊢\left(A\in \mathrm{Fin}\wedge A\in \mathrm{Fin}\right)\to A^A\in \mathrm{Fin}$$
5	4	anidms	$$⊢A\in \mathrm{Fin}\to A^A\in \mathrm{Fin}$$
6	3 5	eqeltrid	$$⊢A\in \mathrm{Fin}\to B\in \mathrm{Fin}$$