Theorem ranrcl4

Description: The first component of a right Kan extension is a functor. (Contributed by Zhi Wang, 4-Nov-2025)

	Ref	Expression
Hypothesis	ranrcl2.l	⊢ ( 𝜑 → 𝐿 ( 𝐹 ( ⟨ 𝐶 , 𝐷 ⟩ Ran 𝐸 ) 𝑋 ) 𝐴 )
Assertion	ranrcl4	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐷 Func 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		ranrcl2.l	⊢ ( 𝜑 → 𝐿 ( 𝐹 ( ⟨ 𝐶 , 𝐷 ⟩ Ran 𝐸 ) 𝑋 ) 𝐴 )
2		eqid	⊢ ( oppCat ‘ ( 𝐷 FuncCat 𝐸 ) ) = ( oppCat ‘ ( 𝐷 FuncCat 𝐸 ) )
3		eqid	⊢ ( oppCat ‘ ( 𝐶 FuncCat 𝐸 ) ) = ( oppCat ‘ ( 𝐶 FuncCat 𝐸 ) )
4	1	ranrcl4lem	⊢ ( 𝜑 → ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) = ⟨ ( 1st ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) , ( 2nd ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) ⟩ )
5	2 3 4 1	isran2	⊢ ( 𝜑 → 𝐿 ( ⟨ ( 1st ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) , tpos ( 2nd ‘ ( ⟨ 𝐷 , 𝐸 ⟩ −∘F 𝐹 ) ) ⟩ ( ( oppCat ‘ ( 𝐷 FuncCat 𝐸 ) ) UP ( oppCat ‘ ( 𝐶 FuncCat 𝐸 ) ) ) 𝑋 ) 𝐴 )
6		eqid	⊢ ( 𝐷 FuncCat 𝐸 ) = ( 𝐷 FuncCat 𝐸 )
7	6	fucbas	⊢ ( 𝐷 Func 𝐸 ) = ( Base ‘ ( 𝐷 FuncCat 𝐸 ) )
8	5 2 7	oppcuprcl4	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐷 Func 𝐸 ) )