Metamath Proof Explorer

Theorem lmdfval2

Description: The set of limits of a diagram. (Contributed by Zhi Wang, 14-Nov-2025)

Ref Expression
Assertion lmdfval2 ( ( 𝐶 Limit 𝐷 ) ‘ 𝐹 ) = ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 )

Proof

Step Hyp Ref Expression
1 lmdfval ⊢ ( 𝐶 Limit 𝐷 ) = ( 𝑓 ∈ ( 𝐷 Func 𝐶 ) ↦ ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝑓 ) )
2 1 mptrcl ⊢ ( 𝑓 ∈ ( ( 𝐶 Limit 𝐷 ) ‘ 𝐹 ) → 𝐹 ∈ ( 𝐷 Func 𝐶 ) )
3 eqid ⊢ ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) = ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) )
4 eqid ⊢ ( 𝐷 FuncCat 𝐶 ) = ( 𝐷 FuncCat 𝐶 )
5 4 fucbas ⊢ ( 𝐷 Func 𝐶 ) = ( Base ‘ ( 𝐷 FuncCat 𝐶 ) )
6 3 5 oppcbas ⊢ ( 𝐷 Func 𝐶 ) = ( Base ‘ ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) )
7 6 uprcl ⊢ ( 𝑓 ∈ ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 ) → ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ∈ ( ( oppCat ‘ 𝐶 ) Func ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) ∧ 𝐹 ∈ ( 𝐷 Func 𝐶 ) ) )
8 7 simprd ⊢ ( 𝑓 ∈ ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 ) → 𝐹 ∈ ( 𝐷 Func 𝐶 ) )
9 oveq2 ⊢ ( 𝑓 = 𝐹 → ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝑓 ) = ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 ) )
10 ovex ⊢ ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 ) ∈ V
11 9 1 10 fvmpt ⊢ ( 𝐹 ∈ ( 𝐷 Func 𝐶 ) → ( ( 𝐶 Limit 𝐷 ) ‘ 𝐹 ) = ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 ) )
12 11 eleq2d ⊢ ( 𝐹 ∈ ( 𝐷 Func 𝐶 ) → ( 𝑓 ∈ ( ( 𝐶 Limit 𝐷 ) ‘ 𝐹 ) ↔ 𝑓 ∈ ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 ) ) )
13 2 8 12 pm5.21nii ⊢ ( 𝑓 ∈ ( ( 𝐶 Limit 𝐷 ) ‘ 𝐹 ) ↔ 𝑓 ∈ ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 ) )
14 13 eqriv ⊢ ( ( 𝐶 Limit 𝐷 ) ‘ 𝐹 ) = ( ( oppFunc ‘ ( 𝐶 Δfunc 𝐷 ) ) ( ( oppCat ‘ 𝐶 ) UP ( oppCat ‘ ( 𝐷 FuncCat 𝐶 ) ) ) 𝐹 )