Theorem relcmd

Description: The set of colimits of a diagram is a relation. (Contributed by Zhi Wang, 13-Nov-2025)

	Ref	Expression
Assertion	relcmd	⊢ Rel ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		relup	⊢ Rel ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 )
2		cmdfval2	⊢ ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 ) = ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 )
3	2	releqi	⊢ ( Rel ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 ) ↔ Rel ( ( 𝐶 Δfunc 𝐷 ) ( 𝐶 UP ( 𝐷 FuncCat 𝐶 ) ) 𝐹 ) )
4	1 3	mpbir	⊢ Rel ( ( 𝐶 Colimit 𝐷 ) ‘ 𝐹 )