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Theorem oicl 7289
Description: The order type of the well-order  R on  A is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Hypothesis
Ref Expression
oicl.1  |-  F  = OrdIso
( R ,  A
)
Assertion
Ref Expression
oicl  |-  Ord  dom  F

Proof of Theorem oicl
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 2316 . . . . 5  |- 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 2316 . . . . 5  |-  { w  e.  A  |  A. j  e.  ran  h  j R w }  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 eqid 2316 . . . . 5  |-  ( 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 7277 . . . 4  |- 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 2316 . . . 4  |-  { 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 . . . 4  |-  F  = OrdIso
( R ,  A
)
7 simpl 443 . . . 4  |-  ( ( R  We  A  /\  R Se  A )  ->  R  We  A )
8 simpr 447 . . . 4  |-  ( ( R  We  A  /\  R Se  A )  ->  R Se  A )
94, 2, 3, 5, 6, 7, 8ordtypelem5 7282 . . 3  |-  ( ( R  We  A  /\  R Se  A )  ->  ( Ord  dom  F  /\  F : dom  F --> A ) )
109simpld 445 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  Ord  dom 
F )
11 ord0 4481 . . 3  |-  Ord  (/)
126oi0 7288 . . . . . 6  |-  ( -.  ( R  We  A  /\  R Se  A )  ->  F  =  (/) )
1312dmeqd 4918 . . . . 5  |-  ( -.  ( R  We  A  /\  R Se  A )  ->  dom  F  =  dom  (/) )
14 dm0 4929 . . . . 5  |-  dom  (/)  =  (/)
1513, 14syl6eq 2364 . . . 4  |-  ( -.  ( R  We  A  /\  R Se  A )  ->  dom  F  =  (/) )
16 ordeq 4436 . . . 4  |-  ( dom 
F  =  (/)  ->  ( Ord  dom  F  <->  Ord  (/) ) )
1715, 16syl 15 . . 3  |-  ( -.  ( R  We  A  /\  R Se  A )  ->  ( Ord  dom  F  <->  Ord  (/) ) )
1811, 17mpbiri 224 . 2  |-  ( -.  ( R  We  A  /\  R Se  A )  ->  Ord  dom  F )
1910, 18pm2.61i 156 1  |-  Ord  dom  F
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1633   A.wral 2577   E.wrex 2578   {crab 2581   _Vcvv 2822   (/)c0 3489   class class class wbr 4060    e. cmpt 4114   Se wse 4387    We wwe 4388   Ord word 4428   Oncon0 4429   dom cdm 4726   ran crn 4727   "cima 4729   -->wf 5288   iota_crio 6339  recscrecs 6429  OrdIsocoi 7269
This theorem is referenced by:  oion  7296  oieu  7299  oismo  7300  oiid  7301  wofib  7305  cantnflt  7418  cantnfp1lem3  7427  cantnflem1b  7433  cantnflem1  7436  wemapwe  7445  cnfcomlem  7447  cnfcom  7448  cnfcom2lem  7449  infxpenlem  7686  hsmexlem1  8097  fpwwe2lem8  8304  fpwwe2lem9  8305  fpwwe2lem10  8306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-riota 6346  df-recs 6430  df-oi 7270
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