MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oiid Unicode version

Theorem oiid 7466
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 4554 . 2  |-  ( Ord 
A  ->  _E  We  A )
2 epse 4525 . . 3  |-  _E Se  A
32a1i 11 . 2  |-  ( Ord 
A  ->  _E Se  A )
4 eqid 2404 . . . . . 6  |- OrdIso (  _E  ,  A )  = OrdIso
(  _E  ,  A
)
54oiiso2 7456 . . . . 5  |-  ( (  _E  We  A  /\  _E Se  A )  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A
) ,  ran OrdIso (  _E  ,  A ) ) )
61, 2, 5sylancl 644 . . . 4  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A ) ,  ran OrdIso (  _E  ,  A ) ) )
7 ordsson 4729 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
84oismo 7465 . . . . . . 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 450 . . . . 5  |-  ( Ord 
A  ->  ran OrdIso (  _E  ,  A )  =  A )
11 isoeq5 6002 . . . . 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 202 . . 3  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A ) ,  A
) )
144oicl 7454 . . . . . 6  |-  Ord  dom OrdIso (  _E  ,  A )
1514a1i 11 . . . . 5  |-  ( Ord 
A  ->  Ord  dom OrdIso (  _E  ,  A ) )
16 id 20 . . . . 5  |-  ( Ord 
A  ->  Ord  A )
17 ordiso2 7440 . . . . 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 1184 . . . 4  |-  ( Ord 
A  ->  dom OrdIso (  _E  ,  A )  =  A )
19 isoeq4 6001 . . . 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 202 . 2  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( A ,  A
) )
22 weniso 6034 . 2  |-  ( (  _E  We  A  /\  _E Se  A  /\ OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( A ,  A
) )  -> OrdIso (  _E  ,  A )  =  (  _I  |`  A ) )
231, 3, 21, 22syl3anc 1184 1  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  =  (  _I  |`  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    C_ wss 3280    _E cep 4452    _I cid 4453   Se wse 4499    We wwe 4500   Ord word 4540   Oncon0 4541   dom cdm 4837   ran crn 4838    |` cres 4839    Isom wiso 5414   Smo wsmo 6566  OrdIsocoi 7434
This theorem is referenced by:  hsmexlem5  8266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-smo 6567  df-recs 6592  df-oi 7435
  Copyright terms: Public domain W3C validator