Description: The order isomorphism of the well-order R on A is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015)
Ref | Expression | ||
---|---|---|---|
Hypothesis | oicl.1 | ⊢ 𝐹 = OrdIso ( 𝑅 , 𝐴 ) | |
Assertion | oiiso | ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴 ) → 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oicl.1 | ⊢ 𝐹 = OrdIso ( 𝑅 , 𝐴 ) | |
2 | exse | ⊢ ( 𝐴 ∈ 𝑉 → 𝑅 Se 𝐴 ) | |
3 | 1 | ordtype | ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) |
4 | 3 | ancoms | ⊢ ( ( 𝑅 Se 𝐴 ∧ 𝑅 We 𝐴 ) → 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) |
5 | 2 4 | sylan | ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴 ) → 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) |