Theorem 1aryenefmnd

Description: The set of unary (endo)functions and the base set of the monoid of endofunctions are equinumerous. (Contributed by AV, 19-May-2024)

	Ref	Expression
Assertion	1aryenefmnd	$$⊢\left(1\mathrm{-aryF}X\right)\approx {\mathrm{Base}}_{\mathrm{EndoFMnd}\left(X\right)}$$

Proof

Step	Hyp	Ref	Expression
1		1aryenef	$$⊢\left(1\mathrm{-aryF}X\right)\approx \left(X^X\right)$$
2		eqid	$$⊢\mathrm{EndoFMnd}\left(X\right)=\mathrm{EndoFMnd}\left(X\right)$$
3		eqid	$$⊢{\mathrm{Base}}_{\mathrm{EndoFMnd}\left(X\right)}={\mathrm{Base}}_{\mathrm{EndoFMnd}\left(X\right)}$$
4	2 3	efmndbas	$$⊢{\mathrm{Base}}_{\mathrm{EndoFMnd}\left(X\right)}=X^X$$
5	1 4	breqtrri	$$⊢\left(1\mathrm{-aryF}X\right)\approx {\mathrm{Base}}_{\mathrm{EndoFMnd}\left(X\right)}$$