Theorem cmdfval2

Description: The set of colimits of a diagram. (Contributed by Zhi Wang, 12-Nov-2025)

	Ref	Expression
Assertion	cmdfval2	⊢ ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 ) = ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		cmdfval	⊢ ( 𝐶 Colimit 𝐷 ) = ( 𝑓 ∈ ( 𝐷 Func 𝐶 ) ↦ ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝑓 ) )
2	1	mptrcl	⊢ ( 𝑓 ∈ ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 ) → 𝐹 ∈ ( 𝐷 Func 𝐶 ) )
3		eqid	⊢ ( 𝐷 FuncCat 𝐶 ) = ( 𝐷 FuncCat 𝐶 )
4	3	fucbas	⊢ ( 𝐷 Func 𝐶 ) = ( Base ‘ ( 𝐷 FuncCat 𝐶 ) )
5	4	uprcl	⊢ ( 𝑓 ∈ ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 ) → ( ( 𝐶 Δfunc 𝐷 ) ∈ ( 𝐶 Func ( 𝐷 FuncCat 𝐶 ) ) ∧ 𝐹 ∈ ( 𝐷 Func 𝐶 ) ) )
6	5	simprd	⊢ ( 𝑓 ∈ ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 ) → 𝐹 ∈ ( 𝐷 Func 𝐶 ) )
7		oveq2	⊢ ( 𝑓 = 𝐹 → ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝑓 ) = ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 ) )
8		ovex	⊢ ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 ) ∈ V
9	7 1 8	fvmpt	⊢ ( 𝐹 ∈ ( 𝐷 Func 𝐶 ) → ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 ) = ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 ) )
10	9	eleq2d	⊢ ( 𝐹 ∈ ( 𝐷 Func 𝐶 ) → ( 𝑓 ∈ ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 ) ↔ 𝑓 ∈ ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 ) ) )
11	2 6 10	pm5.21nii	⊢ ( 𝑓 ∈ ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 ) ↔ 𝑓 ∈ ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 ) )
12	11	eqriv	⊢ ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 ) = ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 )