Theorem fuco11id

Description: The identity morphism of the mapped object. (Contributed by Zhi Wang, 30-Sep-2025)

	Ref	Expression
Hypotheses	fuco11.o	⊢ ( 𝜑 → ( ⟨ 𝐶 , 𝐷 ⟩ ∘F 𝐸 ) = ⟨ 𝑂 , 𝑃 ⟩ )
	fuco11.f	⊢ ( 𝜑 → 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 )
	fuco11.k	⊢ ( 𝜑 → 𝐾 ( 𝐷 Func 𝐸 ) 𝐿 )
	fuco11.u	⊢ ( 𝜑 → 𝑈 = ⟨ ⟨ 𝐾 , 𝐿 ⟩ , ⟨ 𝐹 , 𝐺 ⟩ ⟩ )
	fuco11id.q	⊢ 𝑄 = ( 𝐶 FuncCat 𝐸 )
	fuco11id.i	⊢ 𝐼 = ( Id ‘ 𝑄 )
	fuco11id.1	⊢ 1 = ( Id ‘ 𝐸 )
Assertion	fuco11id	⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑂 ‘ 𝑈 ) ) = ( 1 ∘ ( 𝐾 ∘ 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fuco11.o	⊢ ( 𝜑 → ( ⟨ 𝐶 , 𝐷 ⟩ ∘F 𝐸 ) = ⟨ 𝑂 , 𝑃 ⟩ )
2		fuco11.f	⊢ ( 𝜑 → 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 )
3		fuco11.k	⊢ ( 𝜑 → 𝐾 ( 𝐷 Func 𝐸 ) 𝐿 )
4		fuco11.u	⊢ ( 𝜑 → 𝑈 = ⟨ ⟨ 𝐾 , 𝐿 ⟩ , ⟨ 𝐹 , 𝐺 ⟩ ⟩ )
5		fuco11id.q	⊢ 𝑄 = ( 𝐶 FuncCat 𝐸 )
6		fuco11id.i	⊢ 𝐼 = ( Id ‘ 𝑄 )
7		fuco11id.1	⊢ 1 = ( Id ‘ 𝐸 )
8	1 2 3 4	fuco11cl	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑈 ) ∈ ( 𝐶 Func 𝐸 ) )
9	5 6 7 8	fucid	⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑂 ‘ 𝑈 ) ) = ( 1 ∘ ( 1st ‘ ( 𝑂 ‘ 𝑈 ) ) ) )
10	1 2 3 4	fuco111	⊢ ( 𝜑 → ( 1st ‘ ( 𝑂 ‘ 𝑈 ) ) = ( 𝐾 ∘ 𝐹 ) )
11	10	coeq2d	⊢ ( 𝜑 → ( 1 ∘ ( 1st ‘ ( 𝑂 ‘ 𝑈 ) ) ) = ( 1 ∘ ( 𝐾 ∘ 𝐹 ) ) )
12	9 11	eqtrd	⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑂 ‘ 𝑈 ) ) = ( 1 ∘ ( 𝐾 ∘ 𝐹 ) ) )