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Theorem ordtypelem8 8037
Description: Lemma for ordtype 8044. (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 8033 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
9 fdm 5731 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  dom  O  =  ( T  i^i  dom  F )
)
108, 9syl 17 . . . 4  |-  ( ph  ->  dom  O  =  ( T  i^i  dom  F
) )
11 inss1 3651 . . . . 5  |-  ( T  i^i  dom  F )  C_  T
121, 2, 3, 4, 5, 6, 7ordtypelem2 8031 . . . . . 6  |-  ( ph  ->  Ord  T )
13 ordsson 6613 . . . . . 6  |-  ( Ord 
T  ->  T  C_  On )
1412, 13syl 17 . . . . 5  |-  ( ph  ->  T  C_  On )
1511, 14syl5ss 3442 . . . 4  |-  ( ph  ->  ( T  i^i  dom  F )  C_  On )
1610, 15eqsstrd 3465 . . 3  |-  ( ph  ->  dom  O  C_  On )
17 epweon 6607 . . . 4  |-  _E  We  On
18 weso 4824 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
1917, 18ax-mp 5 . . 3  |-  _E  Or  On
20 soss 4772 . . 3  |-  ( dom 
O  C_  On  ->  (  _E  Or  On  ->  _E  Or  dom  O ) )
2116, 19, 20mpisyl 21 . 2  |-  ( ph  ->  _E  Or  dom  O
)
22 frn 5733 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
238, 22syl 17 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
24 wess 4820 . . . 4  |-  ( ran 
O  C_  A  ->  ( R  We  A  ->  R  We  ran  O ) )
2523, 6, 24sylc 62 . . 3  |-  ( ph  ->  R  We  ran  O
)
26 weso 4824 . . 3  |-  ( R  We  ran  O  ->  R  Or  ran  O )
27 sopo 4771 . . 3  |-  ( R  Or  ran  O  ->  R  Po  ran  O )
2825, 26, 273syl 18 . 2  |-  ( ph  ->  R  Po  ran  O
)
29 ffun 5729 . . . 4  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
308, 29syl 17 . . 3  |-  ( ph  ->  Fun  O )
31 funforn 5798 . . 3  |-  ( Fun 
O  <->  O : dom  O -onto-> ran  O )
3230, 31sylib 200 . 2  |-  ( ph  ->  O : dom  O -onto-> ran  O )
33 epel 4747 . . . . 5  |-  ( a  _E  b  <->  a  e.  b )
341, 2, 3, 4, 5, 6, 7ordtypelem6 8035 . . . . 5  |-  ( (
ph  /\  b  e.  dom  O )  ->  (
a  e.  b  -> 
( O `  a
) R ( O `
 b ) ) )
3533, 34syl5bi 221 . . . 4  |-  ( (
ph  /\  b  e.  dom  O )  ->  (
a  _E  b  -> 
( O `  a
) R ( O `
 b ) ) )
3635ralrimiva 2801 . . 3  |-  ( ph  ->  A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
3736ralrimivw 2802 . 2  |-  ( ph  ->  A. a  e.  dom  O A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
38 soisoi 6217 . 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 1268 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044    i^i cin 3402    C_ wss 3403   class class class wbr 4401    |-> cmpt 4460    _E cep 4742    Po wpo 4752    Or wor 4753   Se wse 4790    We wwe 4791   dom cdm 4833   ran crn 4834   "cima 4836   Ord word 5421   Oncon0 5422   Fun wfun 5575   -->wf 5577   -onto->wfo 5579   ` cfv 5581    Isom wiso 5582   iota_crio 6249  recscrecs 7086  OrdIsocoi 8021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-wrecs 7025  df-recs 7087  df-oi 8022
This theorem is referenced by:  ordtypelem9  8038  ordtypelem10  8039  oiiso2  8043
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