Theorem efmndhash

Description: The monoid of endofunctions on n objects has cardinality n ^ n . (Contributed by AV, 27-Jan-2024)

	Ref	Expression
Hypotheses	efmndbas.g	⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 )
	efmndbas.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	efmndhash	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ 𝐵 ) = ( ( ♯ ‘ 𝐴 ) ↑ ( ♯ ‘ 𝐴 ) ) )

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	3	a1i	⊢ ( 𝐴 ∈ Fin → 𝐵 = ( 𝐴 ↑m 𝐴 ) )
5	4	fveq2d	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ 𝐵 ) = ( ♯ ‘ ( 𝐴 ↑m 𝐴 ) ) )
6		hashmap	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ∈ Fin ) → ( ♯ ‘ ( 𝐴 ↑m 𝐴 ) ) = ( ( ♯ ‘ 𝐴 ) ↑ ( ♯ ‘ 𝐴 ) ) )
7	6	anidms	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ ( 𝐴 ↑m 𝐴 ) ) = ( ( ♯ ‘ 𝐴 ) ↑ ( ♯ ‘ 𝐴 ) ) )
8	5 7	eqtrd	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ 𝐵 ) = ( ( ♯ ‘ 𝐴 ) ↑ ( ♯ ‘ 𝐴 ) ) )