Metamath Proof Explorer


Theorem ielefmnd

Description: The identity function restricted to a set A is an element of the base set of the monoid of endofunctions on A . (Contributed by AV, 27-Jan-2024)

Ref Expression
Hypothesis ielefmnd.g 𝐺 = ( EndoFMnd ‘ 𝐴 )
Assertion ielefmnd ( 𝐴𝑉 → ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 ielefmnd.g 𝐺 = ( EndoFMnd ‘ 𝐴 )
2 f1oi ( I ↾ 𝐴 ) : 𝐴1-1-onto𝐴
3 f1of ( ( I ↾ 𝐴 ) : 𝐴1-1-onto𝐴 → ( I ↾ 𝐴 ) : 𝐴𝐴 )
4 2 3 ax-mp ( I ↾ 𝐴 ) : 𝐴𝐴
5 eqid ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
6 1 5 elefmndbas ( 𝐴𝑉 → ( ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ↔ ( I ↾ 𝐴 ) : 𝐴𝐴 ) )
7 4 6 mpbiri ( 𝐴𝑉 → ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) )