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
|
ordtypelem4 |
|- ( ph -> O : ( T i^i dom F ) --> A ) |
9 |
8
|
fdmd |
|- ( ph -> dom O = ( T i^i dom F ) ) |
10 |
|
inss1 |
|- ( T i^i dom F ) C_ T |
11 |
1 2 3 4 5 6 7
|
ordtypelem2 |
|- ( ph -> Ord T ) |
12 |
|
ordsson |
|- ( Ord T -> T C_ On ) |
13 |
11 12
|
syl |
|- ( ph -> T C_ On ) |
14 |
10 13
|
sstrid |
|- ( ph -> ( T i^i dom F ) C_ On ) |
15 |
9 14
|
eqsstrd |
|- ( ph -> dom O C_ On ) |
16 |
|
epweon |
|- _E We On |
17 |
|
weso |
|- ( _E We On -> _E Or On ) |
18 |
16 17
|
ax-mp |
|- _E Or On |
19 |
|
soss |
|- ( dom O C_ On -> ( _E Or On -> _E Or dom O ) ) |
20 |
15 18 19
|
mpisyl |
|- ( ph -> _E Or dom O ) |
21 |
8
|
frnd |
|- ( ph -> ran O C_ A ) |
22 |
|
wess |
|- ( ran O C_ A -> ( R We A -> R We ran O ) ) |
23 |
21 6 22
|
sylc |
|- ( ph -> R We ran O ) |
24 |
|
weso |
|- ( R We ran O -> R Or ran O ) |
25 |
|
sopo |
|- ( R Or ran O -> R Po ran O ) |
26 |
23 24 25
|
3syl |
|- ( ph -> R Po ran O ) |
27 |
8
|
ffund |
|- ( ph -> Fun O ) |
28 |
|
funforn |
|- ( Fun O <-> O : dom O -onto-> ran O ) |
29 |
27 28
|
sylib |
|- ( ph -> O : dom O -onto-> ran O ) |
30 |
|
epel |
|- ( a _E b <-> a e. b ) |
31 |
1 2 3 4 5 6 7
|
ordtypelem6 |
|- ( ( ph /\ b e. dom O ) -> ( a e. b -> ( O ` a ) R ( O ` b ) ) ) |
32 |
30 31
|
syl5bi |
|- ( ( ph /\ b e. dom O ) -> ( a _E b -> ( O ` a ) R ( O ` b ) ) ) |
33 |
32
|
ralrimiva |
|- ( ph -> A. b e. dom O ( a _E b -> ( O ` a ) R ( O ` b ) ) ) |
34 |
33
|
ralrimivw |
|- ( ph -> A. a e. dom O A. b e. dom O ( a _E b -> ( O ` a ) R ( O ` b ) ) ) |
35 |
|
soisoi |
|- ( ( ( _E Or dom O /\ R Po ran O ) /\ ( O : dom O -onto-> ran O /\ A. a e. dom O A. b e. dom O ( a _E b -> ( O ` a ) R ( O ` b ) ) ) ) -> O Isom _E , R ( dom O , ran O ) ) |
36 |
20 26 29 34 35
|
syl22anc |
|- ( ph -> O Isom _E , R ( dom O , ran O ) ) |