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Theorem smoiso 5858
Description: If  F is an isomorphism from an ordinal onto , which is a subset of the ordinals, then 
F is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
Assertion
Ref Expression
smoiso  F  Isom  _E  ,  _E  ,  Ord  C_  On  Smo  F

Proof of Theorem smoiso
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5390 . . . 4  F 
Isom  _E  ,  _E  ,  F : -1-1-onto->
2 f1of 5069 . . . 4  F : -1-1-onto->  F :
-->
31, 2syl 14 . . 3  F 
Isom  _E  ,  _E  ,  F : -->
4 ffdm 5004 . . . . . 6  F : -->  F : dom  F -->  dom  F  C_
54simpld 105 . . . . 5  F : -->  F : dom  F -->
6 fss 4997 . . . . 5  F : dom  F -->  C_  On  F : dom  F --> On
75, 6sylan 267 . . . 4  F : -->  C_  On 
F : dom  F --> On
873adant2 922 . . 3  F : -->  Ord  C_  On  F : dom  F --> On
93, 8syl3an1 1167 . 2  F  Isom  _E  ,  _E  ,  Ord  C_  On  F : dom  F --> On
10 fdm 4993 . . . . . 6  F : -->  dom 
F
1110eqcomd 2042 . . . . 5  F : -->  dom  F
12 ordeq 4075 . . . . 5  dom  F  Ord  Ord  dom  F
131, 2, 11, 124syl 18 . . . 4  F 
Isom  _E  ,  _E  ,  Ord  Ord  dom  F
1413biimpa 280 . . 3  F  Isom  _E  ,  _E  ,  Ord 
Ord  dom  F
15143adant3 923 . 2  F  Isom  _E  ,  _E  ,  Ord  C_  On  Ord  dom 
F
1610eleq2d 2104 . . . . . . 7  F : -->  dom  F
1710eleq2d 2104 . . . . . . 7  F : -->  dom  F
1816, 17anbi12d 442 . . . . . 6  F : -->  dom  F  dom  F
191, 2, 183syl 17 . . . . 5  F 
Isom  _E  ,  _E  , 
dom  F  dom  F
20 epel 4020 . . . . . . . . 9  _E
21 isorel 5391 . . . . . . . . 9  F  Isom  _E  ,  _E  ,  _E  F `
 _E  F `
2220, 21syl5bbr 183 . . . . . . . 8  F  Isom  _E  ,  _E  ,  F `
 _E  F `
23 ffn 4989 . . . . . . . . . . 11  F : -->  F  Fn
243, 23syl 14 . . . . . . . . . 10  F 
Isom  _E  ,  _E  ,  F  Fn
2524adantr 261 . . . . . . . . 9  F  Isom  _E  ,  _E  ,  F  Fn
26 simprr 484 . . . . . . . . 9  F  Isom  _E  ,  _E  ,
27 funfvex 5135 . . . . . . . . . . 11  Fun  F  dom  F  F `  _V
2827funfni 4942 . . . . . . . . . 10  F  Fn  F `  _V
29 epelg 4018 . . . . . . . . . 10  F `  _V  F `  _E  F `  F `  F `
3028, 29syl 14 . . . . . . . . 9  F  Fn  F `  _E  F `  F `  F `
3125, 26, 30syl2anc 391 . . . . . . . 8  F  Isom  _E  ,  _E  ,  F `  _E  F `  F `  F `
3222, 31bitrd 177 . . . . . . 7  F  Isom  _E  ,  _E  ,  F `
 F `
3332biimpd 132 . . . . . 6  F  Isom  _E  ,  _E  ,  F `  F `
3433ex 108 . . . . 5  F 
Isom  _E  ,  _E  ,  F `  F `
3519, 34sylbid 139 . . . 4  F 
Isom  _E  ,  _E  , 
dom  F  dom  F  F `  F `
3635ralrimivv 2394 . . 3  F 
Isom  _E  ,  _E  ,  dom  F  dom  F  F `  F `
37363ad2ant1 924 . 2  F  Isom  _E  ,  _E  ,  Ord  C_  On  dom  F  dom  F  F `  F `
38 df-smo 5842 . 2  Smo 
F  F : dom  F --> On  Ord  dom  F  dom  F  dom  F  F `  F `
399, 15, 37, 38syl3anbrc 1087 1  F  Isom  _E  ,  _E  ,  Ord  C_  On  Smo  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390  wral 2300   _Vcvv 2551    C_ wss 2911   class class class wbr 3755    _E cep 4015   Ord word 4065   Oncon0 4066   dom cdm 4288    Fn wfn 4840   -->wf 4841   -1-1-onto->wf1o 4844   ` cfv 4845    Isom wiso 4846   Smo wsmo 5841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-f1o 4852  df-fv 4853  df-isom 4854  df-smo 5842
This theorem is referenced by: (None)
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