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Theorem ordtype 7263
Description: For any set-like well-ordered class, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.) (Revised by Mario Carneiro, 25-Jun-2015.)
Hypothesis
Ref Expression
oicl.1  |-  F  = OrdIso
( R ,  A
)
Assertion
Ref Expression
ordtype  |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )

Proof of Theorem ordtype
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 2296 . . 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 2296 . . 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 2296 . . 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 7248 . 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 2296 . 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 443 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  R  We  A )
8 simpr 447 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  R Se  A )
94, 2, 3, 5, 6, 7, 8ordtypelem10 7258 1  |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093    _E cep 4319   Se wse 4366    We wwe 4367   Oncon0 4408   dom cdm 4705   ran crn 4706   "cima 4708    Isom wiso 5272   iota_crio 6313  recscrecs 6403  OrdIsocoi 7240
This theorem is referenced by:  oiexg  7266  oiiso  7268  oieu  7270
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-rep 4147  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-recs 6404  df-oi 7241
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