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 | ⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ielefmnd.g | ⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 ) | |
2 | f1oi | ⊢ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 | |
3 | f1of | ⊢ ( ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 → ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐴 ) | |
4 | 2 3 | ax-mp | ⊢ ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐴 |
5 | eqid | ⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) | |
6 | 1 5 | elefmndbas | ⊢ ( 𝐴 ∈ 𝑉 → ( ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ↔ ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐴 ) ) |
7 | 4 6 | mpbiri | ⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ) |