Metamath Proof Explorer

Theorem par2str

Description: The object set, hom operation, and composition of the concrete two-object parallel-arrow category. (Contributed by Zhi Wang, 14-Sep-2026)

Ref Expression
Hypothesis par2str.c ⊢ C = Base ndx 2 𝑜 Hom ⁡ ndx m ∈ 2 𝑜 , n ∈ 2 𝑜 ⟼ if m = n 2 𝑜 if m = ∅ ∧ n = 1 𝑜 2 𝑜 ∅ comp ⁡ ndx p ∈ 2 𝑜 × 2 𝑜 , q ∈ 2 𝑜 ⟼ r ∈ 3 𝑜 , s ∈ 3 𝑜 ⟼ r ∩ s
Assertion par2str ⊢ 2 𝑜 = Base C ∧ m ∈ 2 𝑜 , n ∈ 2 𝑜 ⟼ if m = n 2 𝑜 if m = ∅ ∧ n = 1 𝑜 2 𝑜 ∅ = Hom ⁡ C ∧ p ∈ 2 𝑜 × 2 𝑜 , q ∈ 2 𝑜 ⟼ r ∈ 3 𝑜 , s ∈ 3 𝑜 ⟼ r ∩ s = comp ⁡ C

Proof

Step Hyp Ref Expression
1 par2str.c ⊢ C = Base ndx 2 𝑜 Hom ⁡ ndx m ∈ 2 𝑜 , n ∈ 2 𝑜 ⟼ if m = n 2 𝑜 if m = ∅ ∧ n = 1 𝑜 2 𝑜 ∅ comp ⁡ ndx p ∈ 2 𝑜 × 2 𝑜 , q ∈ 2 𝑜 ⟼ r ∈ 3 𝑜 , s ∈ 3 𝑜 ⟼ r ∩ s
2 2oex ⊢ 2 𝑜 ∈ V
3 1 2 catbas ⊢ 2 𝑜 = Base C
4 2 2 mpoex ⊢ m ∈ 2 𝑜 , n ∈ 2 𝑜 ⟼ if m = n 2 𝑜 if m = ∅ ∧ n = 1 𝑜 2 𝑜 ∅ ∈ V
5 1 4 cathomfval ⊢ m ∈ 2 𝑜 , n ∈ 2 𝑜 ⟼ if m = n 2 𝑜 if m = ∅ ∧ n = 1 𝑜 2 𝑜 ∅ = Hom ⁡ C
6 2 2 xpex ⊢ 2 𝑜 × 2 𝑜 ∈ V
7 6 2 mpoex ⊢ p ∈ 2 𝑜 × 2 𝑜 , q ∈ 2 𝑜 ⟼ r ∈ 3 𝑜 , s ∈ 3 𝑜 ⟼ r ∩ s ∈ V
8 1 7 catcofval ⊢ p ∈ 2 𝑜 × 2 𝑜 , q ∈ 2 𝑜 ⟼ r ∈ 3 𝑜 , s ∈ 3 𝑜 ⟼ r ∩ s = comp ⁡ C
9 3 5 8 3pm3.2i ⊢ 2 𝑜 = Base C ∧ m ∈ 2 𝑜 , n ∈ 2 𝑜 ⟼ if m = n 2 𝑜 if m = ∅ ∧ n = 1 𝑜 2 𝑜 ∅ = Hom ⁡ C ∧ p ∈ 2 𝑜 × 2 𝑜 , q ∈ 2 𝑜 ⟼ r ∈ 3 𝑜 , s ∈ 3 𝑜 ⟼ r ∩ s = comp ⁡ C