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	⊢ ( 1 -aryF 𝑋 ) ≈ ( Base ‘ ( EndoFMnd ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		1aryenef	⊢ ( 1 -aryF 𝑋 ) ≈ ( 𝑋 ↑m 𝑋 )
2		eqid	⊢ ( EndoFMnd ‘ 𝑋 ) = ( EndoFMnd ‘ 𝑋 )
3		eqid	⊢ ( Base ‘ ( EndoFMnd ‘ 𝑋 ) ) = ( Base ‘ ( EndoFMnd ‘ 𝑋 ) )
4	2 3	efmndbas	⊢ ( Base ‘ ( EndoFMnd ‘ 𝑋 ) ) = ( 𝑋 ↑m 𝑋 )
5	1 4	breqtrri	⊢ ( 1 -aryF 𝑋 ) ≈ ( Base ‘ ( EndoFMnd ‘ 𝑋 ) )