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Theorem oiid 7512
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.)
Assertion
Ref Expression
oiid  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  =  (  _I  |`  A )
)

Proof of Theorem oiid
StepHypRef Expression
1 ordwe 4596 . 2  |-  ( Ord 
A  ->  _E  We  A )
2 epse 4567 . . 3  |-  _E Se  A
32a1i 11 . 2  |-  ( Ord 
A  ->  _E Se  A )
4 eqid 2438 . . . . . 6  |- OrdIso (  _E  ,  A )  = OrdIso
(  _E  ,  A
)
54oiiso2 7502 . . . . 5  |-  ( (  _E  We  A  /\  _E Se  A )  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A
) ,  ran OrdIso (  _E  ,  A ) ) )
61, 2, 5sylancl 645 . . . 4  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A ) ,  ran OrdIso (  _E  ,  A ) ) )
7 ordsson 4772 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
84oismo 7511 . . . . . . 7  |-  ( A 
C_  On  ->  ( Smo OrdIso (  _E  ,  A
)  /\  ran OrdIso (  _E  ,  A )  =  A ) )
97, 8syl 16 . . . . . 6  |-  ( Ord 
A  ->  ( Smo OrdIso (  _E  ,  A )  /\  ran OrdIso (  _E  ,  A )  =  A ) )
109simprd 451 . . . . 5  |-  ( Ord 
A  ->  ran OrdIso (  _E  ,  A )  =  A )
11 isoeq5 6045 . . . . 5  |-  ( 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
) ) )
1210, 11syl 16 . . . 4  |-  ( 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
) ) )
136, 12mpbid 203 . . 3  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A ) ,  A
) )
144oicl 7500 . . . . . 6  |-  Ord  dom OrdIso (  _E  ,  A )
1514a1i 11 . . . . 5  |-  ( Ord 
A  ->  Ord  dom OrdIso (  _E  ,  A ) )
16 id 21 . . . . 5  |-  ( Ord 
A  ->  Ord  A )
17 ordiso2 7486 . . . . 5  |-  ( (OrdIso (  _E  ,  A
)  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A ) ,  A
)  /\  Ord  dom OrdIso (  _E  ,  A )  /\  Ord  A )  ->  dom OrdIso (  _E  ,  A )  =  A )
1813, 15, 16, 17syl3anc 1185 . . . 4  |-  ( Ord 
A  ->  dom OrdIso (  _E  ,  A )  =  A )
19 isoeq4 6044 . . . 4  |-  ( dom OrdIso (  _E  ,  A
)  =  A  -> 
(OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A ) ,  A
)  <-> OrdIso (  _E  ,  A
)  Isom  _E  ,  _E  ( A ,  A ) ) )
2018, 19syl 16 . . 3  |-  ( Ord 
A  ->  (OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A
) ,  A )  <-> OrdIso (  _E  ,  A
)  Isom  _E  ,  _E  ( A ,  A ) ) )
2113, 20mpbid 203 . 2  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( A ,  A
) )
22 weniso 6077 . 2  |-  ( (  _E  We  A  /\  _E Se  A  /\ OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( A ,  A
) )  -> OrdIso (  _E  ,  A )  =  (  _I  |`  A ) )
231, 3, 21, 22syl3anc 1185 1  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  =  (  _I  |`  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    C_ wss 3322    _E cep 4494    _I cid 4495   Se wse 4541    We wwe 4542   Ord word 4582   Oncon0 4583   dom cdm 4880   ran crn 4881    |` cres 4882    Isom wiso 5457   Smo wsmo 6609  OrdIsocoi 7480
This theorem is referenced by:  hsmexlem5  8312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-riota 6551  df-smo 6610  df-recs 6635  df-oi 7481
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