Metamath Proof Explorer

Theorem oiid

Description: The order type of an ordinal under the e. order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015)

Ref Expression
Assertion oiid 
|- ( Ord A -> OrdIso ( _E , A ) = ( _I |` A ) )

Proof

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 C_ On )
8 4 oismo 
 |-  ( A C_ 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 ) )