| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ordtypelem.1 |
|- F = recs ( G ) |
| 2 |
|
ordtypelem.2 |
|- C = { w e. A | A. j e. ran h j R w } |
| 3 |
|
ordtypelem.3 |
|- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) ) |
| 4 |
|
ordtypelem.5 |
|- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t } |
| 5 |
|
ordtypelem.6 |
|- O = OrdIso ( R , A ) |
| 6 |
|
ordtypelem.7 |
|- ( ph -> R We A ) |
| 7 |
|
ordtypelem.8 |
|- ( ph -> R Se A ) |
| 8 |
1 2 3 4 5 6 7
|
ordtypelem2 |
|- ( ph -> Ord T ) |
| 9 |
1
|
tfr1a |
|- ( Fun F /\ Lim dom F ) |
| 10 |
9
|
simpri |
|- Lim dom F |
| 11 |
|
limord |
|- ( Lim dom F -> Ord dom F ) |
| 12 |
10 11
|
ax-mp |
|- Ord dom F |
| 13 |
|
ordin |
|- ( ( Ord T /\ Ord dom F ) -> Ord ( T i^i dom F ) ) |
| 14 |
8 12 13
|
sylancl |
|- ( ph -> Ord ( T i^i dom F ) ) |
| 15 |
1 2 3 4 5 6 7
|
ordtypelem4 |
|- ( ph -> O : ( T i^i dom F ) --> A ) |
| 16 |
15
|
fdmd |
|- ( ph -> dom O = ( T i^i dom F ) ) |
| 17 |
|
ordeq |
|- ( dom O = ( T i^i dom F ) -> ( Ord dom O <-> Ord ( T i^i dom F ) ) ) |
| 18 |
16 17
|
syl |
|- ( ph -> ( Ord dom O <-> Ord ( T i^i dom F ) ) ) |
| 19 |
14 18
|
mpbird |
|- ( ph -> Ord dom O ) |
| 20 |
15
|
ffdmd |
|- ( ph -> O : dom O --> A ) |
| 21 |
19 20
|
jca |
|- ( ph -> ( Ord dom O /\ O : dom O --> A ) ) |