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Theorem oiid 7272
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 4421 . 2  |-  ( Ord 
A  ->  _E  We  A )
2 epse 4392 . . 3  |-  _E Se  A
32a1i 10 . 2  |-  ( Ord 
A  ->  _E Se  A )
4 eqid 2296 . . . . . 6  |- OrdIso (  _E  ,  A )  = OrdIso
(  _E  ,  A
)
54oiiso2 7262 . . . . 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 4597 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
84oismo 7271 . . . . . . 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 5836 . . . . 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 7260 . . . . . 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 7246 . . . . 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 5835 . . . 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 5868 . 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 1632    C_ wss 3165    _E cep 4319    _I cid 4320   Se wse 4366    We wwe 4367   Ord word 4407   Oncon0 4408   dom cdm 4705   ran crn 4706    |` cres 4707    Isom wiso 5272   Smo wsmo 6378  OrdIsocoi 7240
This theorem is referenced by:  hsmexlem5  8072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-smo 6379  df-recs 6404  df-oi 7241
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