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Theorem ordtypelem8 7330
Description: Lemma for ordtype 7337. (Contributed by Mario Carneiro, 17-Oct-2009.)
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 7326 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
9 fdm 5476 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  dom  O  =  ( T  i^i  dom  F )
)
108, 9syl 15 . . . 4  |-  ( ph  ->  dom  O  =  ( T  i^i  dom  F
) )
11 inss1 3465 . . . . 5  |-  ( T  i^i  dom  F )  C_  T
121, 2, 3, 4, 5, 6, 7ordtypelem2 7324 . . . . . 6  |-  ( ph  ->  Ord  T )
13 ordsson 4663 . . . . . 6  |-  ( Ord 
T  ->  T  C_  On )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  T  C_  On )
1511, 14syl5ss 3266 . . . 4  |-  ( ph  ->  ( T  i^i  dom  F )  C_  On )
1610, 15eqsstrd 3288 . . 3  |-  ( ph  ->  dom  O  C_  On )
17 epweon 4657 . . . 4  |-  _E  We  On
18 weso 4466 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
1917, 18ax-mp 8 . . 3  |-  _E  Or  On
20 soss 4414 . . 3  |-  ( dom 
O  C_  On  ->  (  _E  Or  On  ->  _E  Or  dom  O ) )
2116, 19, 20ee10 1376 . 2  |-  ( ph  ->  _E  Or  dom  O
)
22 frn 5478 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
238, 22syl 15 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
24 wess 4462 . . . 4  |-  ( ran 
O  C_  A  ->  ( R  We  A  ->  R  We  ran  O ) )
2523, 6, 24sylc 56 . . 3  |-  ( ph  ->  R  We  ran  O
)
26 weso 4466 . . 3  |-  ( R  We  ran  O  ->  R  Or  ran  O )
27 sopo 4413 . . 3  |-  ( R  Or  ran  O  ->  R  Po  ran  O )
2825, 26, 273syl 18 . 2  |-  ( ph  ->  R  Po  ran  O
)
29 ffun 5474 . . . 4  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
308, 29syl 15 . . 3  |-  ( ph  ->  Fun  O )
31 funforn 5541 . . 3  |-  ( Fun 
O  <->  O : dom  O -onto-> ran  O )
3230, 31sylib 188 . 2  |-  ( ph  ->  O : dom  O -onto-> ran  O )
33 epel 4390 . . . . 5  |-  ( a  _E  b  <->  a  e.  b )
341, 2, 3, 4, 5, 6, 7ordtypelem6 7328 . . . . 5  |-  ( (
ph  /\  b  e.  dom  O )  ->  (
a  e.  b  -> 
( O `  a
) R ( O `
 b ) ) )
3533, 34syl5bi 208 . . . 4  |-  ( (
ph  /\  b  e.  dom  O )  ->  (
a  _E  b  -> 
( O `  a
) R ( O `
 b ) ) )
3635ralrimiva 2702 . . 3  |-  ( ph  ->  A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
3736ralrimivw 2703 . 2  |-  ( ph  ->  A. a  e.  dom  O A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
38 soisoi 5912 . 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 1183 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   {crab 2623   _Vcvv 2864    i^i cin 3227    C_ wss 3228   class class class wbr 4104    e. cmpt 4158    _E cep 4385    Po wpo 4394    Or wor 4395   Se wse 4432    We wwe 4433   Ord word 4473   Oncon0 4474   dom cdm 4771   ran crn 4772   "cima 4774   Fun wfun 5331   -->wf 5333   -onto->wfo 5335   ` cfv 5337    Isom wiso 5338   iota_crio 6384  recscrecs 6474  OrdIsocoi 7314
This theorem is referenced by:  ordtypelem9  7331  ordtypelem10  7332  oiiso2  7336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-riota 6391  df-recs 6475  df-oi 7315
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