Theorem idaf

Description: The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypotheses	idafval.i	⊢ 𝐼 = ( Ida ‘ 𝐶 )
	idafval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	idafval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	idaf.a	⊢ 𝐴 = ( Arrow ‘ 𝐶 )
Assertion	idaf	⊢ ( 𝜑 → 𝐼 : 𝐵 ⟶ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		idafval.i	⊢ 𝐼 = ( Ida ‘ 𝐶 )
2		idafval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		idafval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		idaf.a	⊢ 𝐴 = ( Arrow ‘ 𝐶 )
5		otex	⊢ ⟨ 𝑥 , 𝑥 , ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ⟩ ∈ V
6	5	a1i	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ⟨ 𝑥 , 𝑥 , ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ⟩ ∈ V )
7		eqid	⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 )
8	1 2 3 7	idafval	⊢ ( 𝜑 → 𝐼 = ( 𝑥 ∈ 𝐵 ↦ ⟨ 𝑥 , 𝑥 , ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ⟩ ) )
9		eqid	⊢ ( Homa ‘ 𝐶 ) = ( Homa ‘ 𝐶 )
10	4 9	homarw	⊢ ( 𝑥 ( Homa ‘ 𝐶 ) 𝑥 ) ⊆ 𝐴
11	3	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 ∈ Cat )
12		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 )
13	1 2 11 12 9	idahom	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑥 ) ∈ ( 𝑥 ( Homa ‘ 𝐶 ) 𝑥 ) )
14	10 13	sselid	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑥 ) ∈ 𝐴 )
15	6 8 14	fmpt2d	⊢ ( 𝜑 → 𝐼 : 𝐵 ⟶ 𝐴 )