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 ∈ Fin → 𝐵 ∈ 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 ∈ Fin ∧ 𝐴 ∈ Fin ) → ( 𝐴 ↑m 𝐴 ) ∈ Fin )
5	4	anidms	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ↑m 𝐴 ) ∈ Fin )
6	3 5	eqeltrid	⊢ ( 𝐴 ∈ Fin → 𝐵 ∈ Fin )