Theorem prcof2a

Description: The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025)

	Ref	Expression
Hypotheses	prcof2a.n	⊢ 𝑁 = ( 𝐷 Nat 𝐸 )
	prcof2a.k	⊢ ( 𝜑 → 𝐾 ∈ ( 𝐷 Func 𝐸 ) )
	prcof2a.l	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐷 Func 𝐸 ) )
	prcof2a.p	⊢ ( 𝜑 → ( 2nd ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) = 𝑃 )
	prcof2a.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑈 )
Assertion	prcof2a	⊢ ( 𝜑 → ( 𝐾 𝑃 𝐿 ) = ( 𝑎 ∈ ( 𝐾 𝑁 𝐿 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		prcof2a.n	⊢ 𝑁 = ( 𝐷 Nat 𝐸 )
2		prcof2a.k	⊢ ( 𝜑 → 𝐾 ∈ ( 𝐷 Func 𝐸 ) )
3		prcof2a.l	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐷 Func 𝐸 ) )
4		prcof2a.p	⊢ ( 𝜑 → ( 2nd ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) = 𝑃 )
5		prcof2a.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑈 )
6		eqid	⊢ ( 𝐷 Func 𝐸 ) = ( 𝐷 Func 𝐸 )
7	2	func1st2nd	⊢ ( 𝜑 → ( 1st ‘ 𝐾 ) ( 𝐷 Func 𝐸 ) ( 2nd ‘ 𝐾 ) )
8	7	funcrcl2	⊢ ( 𝜑 → 𝐷 ∈ Cat )
9	7	funcrcl3	⊢ ( 𝜑 → 𝐸 ∈ Cat )
10	6 1 8 9 5	prcofvala	⊢ ( 𝜑 → ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) = ⟨ ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑘 ∘func 𝐹 ) ) , ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) , 𝑙 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑎 ∈ ( 𝑘 𝑁 𝑙 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) ) ⟩ )
11	10	fveq2d	⊢ ( 𝜑 → ( 2nd ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) = ( 2nd ‘ ⟨ ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑘 ∘func 𝐹 ) ) , ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) , 𝑙 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑎 ∈ ( 𝑘 𝑁 𝑙 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) ) ⟩ ) )
12		ovex	⊢ ( 𝐷 Func 𝐸 ) ∈ V
13	12	mptex	⊢ ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑘 ∘func 𝐹 ) ) ∈ V
14	12 12	mpoex	⊢ ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) , 𝑙 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑎 ∈ ( 𝑘 𝑁 𝑙 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) ) ∈ V
15	13 14	op2nd	⊢ ( 2nd ‘ ⟨ ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑘 ∘func 𝐹 ) ) , ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) , 𝑙 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑎 ∈ ( 𝑘 𝑁 𝑙 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) ) ⟩ ) = ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) , 𝑙 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑎 ∈ ( 𝑘 𝑁 𝑙 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) )
16	11 15	eqtrdi	⊢ ( 𝜑 → ( 2nd ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) = ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) , 𝑙 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑎 ∈ ( 𝑘 𝑁 𝑙 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) ) )
17	4 16	eqtr3d	⊢ ( 𝜑 → 𝑃 = ( 𝑘 ∈ ( 𝐷 Func 𝐸 ) , 𝑙 ∈ ( 𝐷 Func 𝐸 ) ↦ ( 𝑎 ∈ ( 𝑘 𝑁 𝑙 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) ) )
18		simprl	⊢ ( ( 𝜑 ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → 𝑘 = 𝐾 )
19		simprr	⊢ ( ( 𝜑 ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → 𝑙 = 𝐿 )
20	18 19	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → ( 𝑘 𝑁 𝑙 ) = ( 𝐾 𝑁 𝐿 ) )
21	20	mpteq1d	⊢ ( ( 𝜑 ∧ ( 𝑘 = 𝐾 ∧ 𝑙 = 𝐿 ) ) → ( 𝑎 ∈ ( 𝑘 𝑁 𝑙 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) = ( 𝑎 ∈ ( 𝐾 𝑁 𝐿 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) )
22		ovex	⊢ ( 𝐾 𝑁 𝐿 ) ∈ V
23	22	mptex	⊢ ( 𝑎 ∈ ( 𝐾 𝑁 𝐿 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) ∈ V
24	23	a1i	⊢ ( 𝜑 → ( 𝑎 ∈ ( 𝐾 𝑁 𝐿 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) ∈ V )
25	17 21 2 3 24	ovmpod	⊢ ( 𝜑 → ( 𝐾 𝑃 𝐿 ) = ( 𝑎 ∈ ( 𝐾 𝑁 𝐿 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) )