| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ordwe |
⊢ ( Ord 𝐴 → E We 𝐴 ) |
| 2 |
|
epse |
⊢ E Se 𝐴 |
| 3 |
2
|
a1i |
⊢ ( Ord 𝐴 → E Se 𝐴 ) |
| 4 |
|
eqid |
⊢ OrdIso ( E , 𝐴 ) = OrdIso ( E , 𝐴 ) |
| 5 |
4
|
oiiso2 |
⊢ ( ( E We 𝐴 ∧ E Se 𝐴 ) → OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , ran OrdIso ( E , 𝐴 ) ) ) |
| 6 |
1 2 5
|
sylancl |
⊢ ( Ord 𝐴 → OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , ran OrdIso ( E , 𝐴 ) ) ) |
| 7 |
|
ordsson |
⊢ ( Ord 𝐴 → 𝐴 ⊆ On ) |
| 8 |
4
|
oismo |
⊢ ( 𝐴 ⊆ On → ( Smo OrdIso ( E , 𝐴 ) ∧ ran OrdIso ( E , 𝐴 ) = 𝐴 ) ) |
| 9 |
7 8
|
syl |
⊢ ( Ord 𝐴 → ( Smo OrdIso ( E , 𝐴 ) ∧ ran OrdIso ( E , 𝐴 ) = 𝐴 ) ) |
| 10 |
|
isoeq5 |
⊢ ( ran OrdIso ( E , 𝐴 ) = 𝐴 → ( OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , ran OrdIso ( E , 𝐴 ) ) ↔ OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , 𝐴 ) ) ) |
| 11 |
9 10
|
simpl2im |
⊢ ( Ord 𝐴 → ( OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , ran OrdIso ( E , 𝐴 ) ) ↔ OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , 𝐴 ) ) ) |
| 12 |
6 11
|
mpbid |
⊢ ( Ord 𝐴 → OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , 𝐴 ) ) |
| 13 |
4
|
oicl |
⊢ Ord dom OrdIso ( E , 𝐴 ) |
| 14 |
13
|
a1i |
⊢ ( Ord 𝐴 → Ord dom OrdIso ( E , 𝐴 ) ) |
| 15 |
|
id |
⊢ ( Ord 𝐴 → Ord 𝐴 ) |
| 16 |
|
ordiso2 |
⊢ ( ( OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , 𝐴 ) ∧ Ord dom OrdIso ( E , 𝐴 ) ∧ Ord 𝐴 ) → dom OrdIso ( E , 𝐴 ) = 𝐴 ) |
| 17 |
12 14 15 16
|
syl3anc |
⊢ ( Ord 𝐴 → dom OrdIso ( E , 𝐴 ) = 𝐴 ) |
| 18 |
|
isoeq4 |
⊢ ( dom OrdIso ( E , 𝐴 ) = 𝐴 → ( OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , 𝐴 ) ↔ OrdIso ( E , 𝐴 ) Isom E , E ( 𝐴 , 𝐴 ) ) ) |
| 19 |
17 18
|
syl |
⊢ ( Ord 𝐴 → ( OrdIso ( E , 𝐴 ) Isom E , E ( dom OrdIso ( E , 𝐴 ) , 𝐴 ) ↔ OrdIso ( E , 𝐴 ) Isom E , E ( 𝐴 , 𝐴 ) ) ) |
| 20 |
12 19
|
mpbid |
⊢ ( Ord 𝐴 → OrdIso ( E , 𝐴 ) Isom E , E ( 𝐴 , 𝐴 ) ) |
| 21 |
|
weniso |
⊢ ( ( E We 𝐴 ∧ E Se 𝐴 ∧ OrdIso ( E , 𝐴 ) Isom E , E ( 𝐴 , 𝐴 ) ) → OrdIso ( E , 𝐴 ) = ( I ↾ 𝐴 ) ) |
| 22 |
1 3 20 21
|
syl3anc |
⊢ ( Ord 𝐴 → OrdIso ( E , 𝐴 ) = ( I ↾ 𝐴 ) ) |