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Metamath Proof Explorer

Theorem soinxp

Description: Intersection of total order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014)

Ref Expression
Assertion soinxp ( 𝑅 Or 𝐴 ↔ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Or 𝐴 )

Proof

Step Hyp Ref Expression
1 poinxp ⊢ ( 𝑅 Po 𝐴 ↔ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Po 𝐴 )
2 brinxp ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 ↔ 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑦 ) )
3 biidd ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑦 ) )
4 brinxp ⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 𝑅 𝑥 ↔ 𝑦 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ) )
5 4 ancoms ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑦 𝑅 𝑥 ↔ 𝑦 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ) )
6 2 3 5 3orbi123d ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ) ) )
7 6 ralbidva ⊢ ( 𝑥 ∈ 𝐴 → ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ) ) )
8 7 ralbiia ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ) )
9 1 8 anbi12i ⊢ ( ( 𝑅 Po 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) ↔ ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Po 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ) ) )
10 df-so ⊢ ( 𝑅 Or 𝐴 ↔ ( 𝑅 Po 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) )
11 df-so ⊢ ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Or 𝐴 ↔ ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Po 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ) ) )
12 9 10 11 3bitr4i ⊢ ( 𝑅 Or 𝐴 ↔ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Or 𝐴 )