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 ) )