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

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	3	a1i	$$⊢A\in \mathrm{Fin}\to B=A^A$$
5	4	fveq2d	$$⊢A\in \mathrm{Fin}\to \left|B\right|=\left|A^A\right|$$
6		hashmap	$$⊢\left(A\in \mathrm{Fin}\wedge A\in \mathrm{Fin}\right)\to \left|A^A\right|={\left|A\right|}^{\left|A\right|}$$
7	6	anidms	$$⊢A\in \mathrm{Fin}\to \left|A^A\right|={\left|A\right|}^{\left|A\right|}$$
8	5 7	eqtrd	$$⊢A\in \mathrm{Fin}\to \left|B\right|={\left|A\right|}^{\left|A\right|}$$