Theorem prcof21a

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

	Ref	Expression
Hypotheses	prcof21a.n	⊢ 𝑁 = ( 𝐷 Nat 𝐸 )
	prcof21a.a	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐾 𝑁 𝐿 ) )
	prcof21a.p	⊢ ( 𝜑 → ( 2nd ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) = 𝑃 )
	prcof21a.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑈 )
Assertion	prcof21a	⊢ ( 𝜑 → ( ( 𝐾 𝑃 𝐿 ) ‘ 𝐴 ) = ( 𝐴 ∘ ( 1st ‘ 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		prcof21a.n	⊢ 𝑁 = ( 𝐷 Nat 𝐸 )
2		prcof21a.a	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐾 𝑁 𝐿 ) )
3		prcof21a.p	⊢ ( 𝜑 → ( 2nd ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) = 𝑃 )
4		prcof21a.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑈 )
5	1	natrcl	⊢ ( 𝐴 ∈ ( 𝐾 𝑁 𝐿 ) → ( 𝐾 ∈ ( 𝐷 Func 𝐸 ) ∧ 𝐿 ∈ ( 𝐷 Func 𝐸 ) ) )
6	2 5	syl	⊢ ( 𝜑 → ( 𝐾 ∈ ( 𝐷 Func 𝐸 ) ∧ 𝐿 ∈ ( 𝐷 Func 𝐸 ) ) )
7	6	simpld	⊢ ( 𝜑 → 𝐾 ∈ ( 𝐷 Func 𝐸 ) )
8	6	simprd	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐷 Func 𝐸 ) )
9	1 7 8 3 4	prcof2a	⊢ ( 𝜑 → ( 𝐾 𝑃 𝐿 ) = ( 𝑎 ∈ ( 𝐾 𝑁 𝐿 ) ↦ ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) ) )
10		simpr	⊢ ( ( 𝜑 ∧ 𝑎 = 𝐴 ) → 𝑎 = 𝐴 )
11	10	coeq1d	⊢ ( ( 𝜑 ∧ 𝑎 = 𝐴 ) → ( 𝑎 ∘ ( 1st ‘ 𝐹 ) ) = ( 𝐴 ∘ ( 1st ‘ 𝐹 ) ) )
12		fvexd	⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) ∈ V )
13	2 12	coexd	⊢ ( 𝜑 → ( 𝐴 ∘ ( 1st ‘ 𝐹 ) ) ∈ V )
14	9 11 2 13	fvmptd	⊢ ( 𝜑 → ( ( 𝐾 𝑃 𝐿 ) ‘ 𝐴 ) = ( 𝐴 ∘ ( 1st ‘ 𝐹 ) ) )