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Theorem oiid 7256
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 4405 . 2  |-  ( Ord 
A  ->  _E  We  A )
2 epse 4376 . . 3  |-  _E Se  A
32a1i 10 . 2  |-  ( Ord 
A  ->  _E Se  A )
4 eqid 2283 . . . . . 6  |- OrdIso (  _E  ,  A )  = OrdIso
(  _E  ,  A
)
54oiiso2 7246 . . . . 5  |-  ( (  _E  We  A  /\  _E Se  A )  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A
) ,  ran OrdIso (  _E  ,  A ) ) )
61, 2, 5sylancl 643 . . . 4  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A ) ,  ran OrdIso (  _E  ,  A ) ) )
7 ordsson 4581 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
84oismo 7255 . . . . . . 7  |-  ( A 
C_  On  ->  ( Smo OrdIso (  _E  ,  A
)  /\  ran OrdIso (  _E  ,  A )  =  A ) )
97, 8syl 15 . . . . . 6  |-  ( Ord 
A  ->  ( Smo OrdIso (  _E  ,  A )  /\  ran OrdIso (  _E  ,  A )  =  A ) )
109simprd 449 . . . . 5  |-  ( Ord 
A  ->  ran OrdIso (  _E  ,  A )  =  A )
11 isoeq5 5820 . . . . 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 15 . . . 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 201 . . 3  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A ) ,  A
) )
144oicl 7244 . . . . . 6  |-  Ord  dom OrdIso (  _E  ,  A )
1514a1i 10 . . . . 5  |-  ( Ord 
A  ->  Ord  dom OrdIso (  _E  ,  A ) )
16 id 19 . . . . 5  |-  ( Ord 
A  ->  Ord  A )
17 ordiso2 7230 . . . . 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 1182 . . . 4  |-  ( Ord 
A  ->  dom OrdIso (  _E  ,  A )  =  A )
19 isoeq4 5819 . . . 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 15 . . 3  |-  ( Ord 
A  ->  (OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  A
) ,  A )  <-> OrdIso (  _E  ,  A
)  Isom  _E  ,  _E  ( A ,  A ) ) )
2113, 20mpbid 201 . 2  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( A ,  A
) )
22 weniso 5852 . 2  |-  ( (  _E  We  A  /\  _E Se  A  /\ OrdIso (  _E  ,  A )  Isom  _E  ,  _E  ( A ,  A
) )  -> OrdIso (  _E  ,  A )  =  (  _I  |`  A ) )
231, 3, 21, 22syl3anc 1182 1  |-  ( Ord 
A  -> OrdIso (  _E  ,  A )  =  (  _I  |`  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    C_ wss 3152    _E cep 4303    _I cid 4304   Se wse 4350    We wwe 4351   Ord word 4391   Oncon0 4392   dom cdm 4689   ran crn 4690    |` cres 4691    Isom wiso 5256   Smo wsmo 6362  OrdIsocoi 7224
This theorem is referenced by:  hsmexlem5  8056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-smo 6363  df-recs 6388  df-oi 7225
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