Metamath Proof Explorer

Theorem ordtypelem8

Description: Lemma for ordtype . (Contributed by Mario Carneiro, 25-Jun-2015)

Ref Expression
Hypotheses ordtypelem.1 
|- F = recs ( G )
ordtypelem.2 
|- C = { w e. A | A. j e. ran h j R w }
ordtypelem.3 
|- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) )
ordtypelem.5 
|- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t }
ordtypelem.6 
|- O = OrdIso ( R , A )
ordtypelem.7 
|- ( ph -> R We A )
ordtypelem.8 
|- ( ph -> R Se A )
Assertion ordtypelem8 
|- ( ph -> O Isom _E , R ( dom O , ran O ) )

Proof

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 biimtrid 
 |-  ( ( 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 ) )