Theorem natrcl3

Description: Reverse closure for a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025)

	Ref	Expression
Hypotheses	natrcl2.n	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
	natrcl2.a	⊢ ( 𝜑 → 𝐴 ∈ ( ⟨ 𝐹 , 𝐺 ⟩ 𝑁 ⟨ 𝐾 , 𝐿 ⟩ ) )
Assertion	natrcl3	⊢ ( 𝜑 → 𝐾 ( 𝐶 Func 𝐷 ) 𝐿 )

Proof

Step	Hyp	Ref	Expression
1		natrcl2.n	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
2		natrcl2.a	⊢ ( 𝜑 → 𝐴 ∈ ( ⟨ 𝐹 , 𝐺 ⟩ 𝑁 ⟨ 𝐾 , 𝐿 ⟩ ) )
3	1	natrcl	⊢ ( 𝐴 ∈ ( ⟨ 𝐹 , 𝐺 ⟩ 𝑁 ⟨ 𝐾 , 𝐿 ⟩ ) → ( ⟨ 𝐹 , 𝐺 ⟩ ∈ ( 𝐶 Func 𝐷 ) ∧ ⟨ 𝐾 , 𝐿 ⟩ ∈ ( 𝐶 Func 𝐷 ) ) )
4	2 3	syl	⊢ ( 𝜑 → ( ⟨ 𝐹 , 𝐺 ⟩ ∈ ( 𝐶 Func 𝐷 ) ∧ ⟨ 𝐾 , 𝐿 ⟩ ∈ ( 𝐶 Func 𝐷 ) ) )
5	4	simprd	⊢ ( 𝜑 → ⟨ 𝐾 , 𝐿 ⟩ ∈ ( 𝐶 Func 𝐷 ) )
6		df-br	⊢ ( 𝐾 ( 𝐶 Func 𝐷 ) 𝐿 ↔ ⟨ 𝐾 , 𝐿 ⟩ ∈ ( 𝐶 Func 𝐷 ) )
7	5 6	sylibr	⊢ ( 𝜑 → 𝐾 ( 𝐶 Func 𝐷 ) 𝐿 )