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Metamath Proof Explorer

Theorem efmndid

Description: The identity function restricted to a set A is the identity element of the monoid of endofunctions on A . (Contributed by AV, 25-Jan-2024)

Ref Expression
Hypothesis ielefmnd.g ⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 )
Assertion efmndid ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) = ( 0g ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 ielefmnd.g ⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 )
2 eqid ⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
3 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
4 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
5 1 ielefmnd ⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) )
6 1 2 4 efmndov ⊢ ( ( ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( ( I ↾ 𝐴 ) ( +g ‘ 𝐺 ) 𝑓 ) = ( ( I ↾ 𝐴 ) ∘ 𝑓 ) )
7 5 6 sylan ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( ( I ↾ 𝐴 ) ( +g ‘ 𝐺 ) 𝑓 ) = ( ( I ↾ 𝐴 ) ∘ 𝑓 ) )
8 1 2 efmndbasf ⊢ ( 𝑓 ∈ ( Base ‘ 𝐺 ) → 𝑓 : 𝐴 ⟶ 𝐴 )
9 8 adantl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → 𝑓 : 𝐴 ⟶ 𝐴 )
10 fcoi2 ⊢ ( 𝑓 : 𝐴 ⟶ 𝐴 → ( ( I ↾ 𝐴 ) ∘ 𝑓 ) = 𝑓 )
11 9 10 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( ( I ↾ 𝐴 ) ∘ 𝑓 ) = 𝑓 )
12 7 11 eqtrd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( ( I ↾ 𝐴 ) ( +g ‘ 𝐺 ) 𝑓 ) = 𝑓 )
13 5 anim1ci ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ∈ ( Base ‘ 𝐺 ) ∧ ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ) )
14 1 2 4 efmndov ⊢ ( ( 𝑓 ∈ ( Base ‘ 𝐺 ) ∧ ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ( +g ‘ 𝐺 ) ( I ↾ 𝐴 ) ) = ( 𝑓 ∘ ( I ↾ 𝐴 ) ) )
15 13 14 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ( +g ‘ 𝐺 ) ( I ↾ 𝐴 ) ) = ( 𝑓 ∘ ( I ↾ 𝐴 ) ) )
16 fcoi1 ⊢ ( 𝑓 : 𝐴 ⟶ 𝐴 → ( 𝑓 ∘ ( I ↾ 𝐴 ) ) = 𝑓 )
17 9 16 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ∘ ( I ↾ 𝐴 ) ) = 𝑓 )
18 15 17 eqtrd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ( +g ‘ 𝐺 ) ( I ↾ 𝐴 ) ) = 𝑓 )
19 2 3 4 5 12 18 ismgmid2 ⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) = ( 0g ‘ 𝐺 ) )