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Theorem oiiso2 7456
Description: The order isomorphism of the well-order  R on  A is an isomorphism onto  ran  O (which is a subset of  A by oif 7455). (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypothesis
Ref Expression
oicl.1  |-  F  = OrdIso
( R ,  A
)
Assertion
Ref Expression
oiiso2  |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  ran  F ) )

Proof of Theorem oiiso2
Dummy variables  u  t  v  x  h  j  w  z  f 
i  r  s  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |- recs ( ( h  e.  _V  |->  (
iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  = recs ( ( h  e.  _V  |->  (
iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
2 eqid 2404 . . 3  |-  { w  e.  A  |  A. j  e.  ran  h  j R w }  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 eqid 2404 . . 3  |-  ( h  e.  _V  |->  ( iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) )  =  ( h  e.  _V  |->  ( iota_ v  e.  {
w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) )
41, 2, 3ordtypecbv 7442 . 2  |- recs ( ( f  e.  _V  |->  (
iota_ s  e.  { y  e.  A  |  A. i  e.  ran  f  i R y } A. r  e.  { y  e.  A  |  A. i  e.  ran  f  i R y }  -.  r R s ) ) )  = recs ( ( h  e.  _V  |->  (
iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
5 eqid 2404 . 2  |-  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( f  e.  _V  |->  ( iota_ s  e.  { y  e.  A  |  A. i  e.  ran  f  i R y } A. r  e.  { y  e.  A  |  A. i  e.  ran  f  i R y }  -.  r R s ) ) )
" x ) z R t }  =  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( f  e. 
_V  |->  ( iota_ s  e. 
{ y  e.  A  |  A. i  e.  ran  f  i R y } A. r  e. 
{ y  e.  A  |  A. i  e.  ran  f  i R y }  -.  r R s ) ) )
" x ) z R t }
6 oicl.1 . 2  |-  F  = OrdIso
( R ,  A
)
7 simpl 444 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  R  We  A )
8 simpr 448 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  R Se  A )
94, 2, 3, 5, 6, 7, 8ordtypelem8 7450 1  |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  ran  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226    _E cep 4452   Se wse 4499    We wwe 4500   Oncon0 4541   dom cdm 4837   ran crn 4838   "cima 4840    Isom wiso 5414   iota_crio 6501  recscrecs 6591  OrdIsocoi 7434
This theorem is referenced by:  oismo  7465  oiid  7466  hsmexlem1  8262
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-recs 6592  df-oi 7435
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