Theorem fuco112xa

Description: The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly for the morphism part of the composed functor. A morphism is mapped by two functors in succession. (Contributed by Zhi Wang, 3-Oct-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 ⟩ ⟩ )
	fuco111x.x	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
	fuco112x.y	⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) )
	fuco112xa.a	⊢ ( 𝜑 → 𝐴 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
Assertion	fuco112xa	⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ ( 𝑂 ‘ 𝑈 ) ) 𝑌 ) ‘ 𝐴 ) = ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑌 ) ) ‘ ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐴 ) ) )

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		fuco111x.x	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
6		fuco112x.y	⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) )
7		fuco112xa.a	⊢ ( 𝜑 → 𝐴 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
8	1 2 3 4 5 6	fuco112x	⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ ( 𝑂 ‘ 𝑈 ) ) 𝑌 ) = ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑌 ) ) ∘ ( 𝑋 𝐺 𝑌 ) ) )
9	8	fveq1d	⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ ( 𝑂 ‘ 𝑈 ) ) 𝑌 ) ‘ 𝐴 ) = ( ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑌 ) ) ∘ ( 𝑋 𝐺 𝑌 ) ) ‘ 𝐴 ) )
10		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
11		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
12		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
13	10 11 12 2 5 6	funcf2	⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ⟶ ( ( 𝐹 ‘ 𝑋 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑌 ) ) )
14	13 7	fvco3d	⊢ ( 𝜑 → ( ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑌 ) ) ∘ ( 𝑋 𝐺 𝑌 ) ) ‘ 𝐴 ) = ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑌 ) ) ‘ ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐴 ) ) )
15	9 14	eqtrd	⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ ( 𝑂 ‘ 𝑈 ) ) 𝑌 ) ‘ 𝐴 ) = ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑌 ) ) ‘ ( ( 𝑋 𝐺 𝑌 ) ‘ 𝐴 ) ) )