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Theorem ordtypelem8 7962
Description: Lemma for ordtype 7969. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1  |-  F  = recs ( G )
ordtypelem.2  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
ordtypelem.3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
ordtypelem.5  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
ordtypelem.6  |-  O  = OrdIso
( R ,  A
)
ordtypelem.7  |-  ( ph  ->  R  We  A )
ordtypelem.8  |-  ( ph  ->  R Se  A )
Assertion
Ref Expression
ordtypelem8  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
Distinct variable groups:    v, u, C    h, j, t, u, v, w, x, z, R    A, h, j, t, u, v, w, x, z    t, O, u, v, x    ph, t, x    h, F, j, t, u, v, w, x, z
Allowed substitution hints:    ph( z, w, v, u, h, j)    C( x, z, w, t, h, j)    T( x, z, w, v, u, t, h, j)    G( x, z, w, v, u, t, h, j)    O( z, w, h, j)

Proof of Theorem ordtypelem8
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . . . . 6  |-  F  = recs ( G )
2 ordtypelem.2 . . . . . 6  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 ordtypelem.3 . . . . . 6  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
4 ordtypelem.5 . . . . . 6  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
5 ordtypelem.6 . . . . . 6  |-  O  = OrdIso
( R ,  A
)
6 ordtypelem.7 . . . . . 6  |-  ( ph  ->  R  We  A )
7 ordtypelem.8 . . . . . 6  |-  ( ph  ->  R Se  A )
81, 2, 3, 4, 5, 6, 7ordtypelem4 7958 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
9 fdm 5741 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  dom  O  =  ( T  i^i  dom  F )
)
108, 9syl 16 . . . 4  |-  ( ph  ->  dom  O  =  ( T  i^i  dom  F
) )
11 inss1 3723 . . . . 5  |-  ( T  i^i  dom  F )  C_  T
121, 2, 3, 4, 5, 6, 7ordtypelem2 7956 . . . . . 6  |-  ( ph  ->  Ord  T )
13 ordsson 6620 . . . . . 6  |-  ( Ord 
T  ->  T  C_  On )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  T  C_  On )
1511, 14syl5ss 3520 . . . 4  |-  ( ph  ->  ( T  i^i  dom  F )  C_  On )
1610, 15eqsstrd 3543 . . 3  |-  ( ph  ->  dom  O  C_  On )
17 epweon 6614 . . . 4  |-  _E  We  On
18 weso 4876 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
1917, 18ax-mp 5 . . 3  |-  _E  Or  On
20 soss 4824 . . 3  |-  ( dom 
O  C_  On  ->  (  _E  Or  On  ->  _E  Or  dom  O ) )
2116, 19, 20mpisyl 18 . 2  |-  ( ph  ->  _E  Or  dom  O
)
22 frn 5743 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
238, 22syl 16 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
24 wess 4872 . . . 4  |-  ( ran 
O  C_  A  ->  ( R  We  A  ->  R  We  ran  O ) )
2523, 6, 24sylc 60 . . 3  |-  ( ph  ->  R  We  ran  O
)
26 weso 4876 . . 3  |-  ( R  We  ran  O  ->  R  Or  ran  O )
27 sopo 4823 . . 3  |-  ( R  Or  ran  O  ->  R  Po  ran  O )
2825, 26, 273syl 20 . 2  |-  ( ph  ->  R  Po  ran  O
)
29 ffun 5739 . . . 4  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
308, 29syl 16 . . 3  |-  ( ph  ->  Fun  O )
31 funforn 5808 . . 3  |-  ( Fun 
O  <->  O : dom  O -onto-> ran  O )
3230, 31sylib 196 . 2  |-  ( ph  ->  O : dom  O -onto-> ran  O )
33 epel 4800 . . . . 5  |-  ( a  _E  b  <->  a  e.  b )
341, 2, 3, 4, 5, 6, 7ordtypelem6 7960 . . . . 5  |-  ( (
ph  /\  b  e.  dom  O )  ->  (
a  e.  b  -> 
( O `  a
) R ( O `
 b ) ) )
3533, 34syl5bi 217 . . . 4  |-  ( (
ph  /\  b  e.  dom  O )  ->  (
a  _E  b  -> 
( O `  a
) R ( O `
 b ) ) )
3635ralrimiva 2881 . . 3  |-  ( ph  ->  A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
3736ralrimivw 2882 . 2  |-  ( ph  ->  A. a  e.  dom  O A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
38 soisoi 6223 . 2  |-  ( ( (  _E  Or  dom  O  /\  R  Po  ran  O )  /\  ( O : dom  O -onto-> ran  O  /\  A. a  e. 
dom  O A. b  e.  dom  O ( a  _E  b  ->  ( O `  a ) R ( O `  b ) ) ) )  ->  O  Isom  _E  ,  R  ( dom 
O ,  ran  O
) )
3921, 28, 32, 37, 38syl22anc 1229 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118    i^i cin 3480    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511    _E cep 4795    Po wpo 4804    Or wor 4805   Se wse 4842    We wwe 4843   Ord word 4883   Oncon0 4884   dom cdm 5005   ran crn 5006   "cima 5008   Fun wfun 5588   -->wf 5590   -onto->wfo 5592   ` cfv 5594    Isom wiso 5595   iota_crio 6255  recscrecs 7053  OrdIsocoi 7946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-recs 7054  df-oi 7947
This theorem is referenced by:  ordtypelem9  7963  ordtypelem10  7964  oiiso2  7968
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