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Theorem ordtypelem8 7240
Description: Lemma for ordtype 7247. (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 7236 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
9 fdm 5393 . . . . 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 3389 . . . . 5  |-  ( T  i^i  dom  F )  C_  T
121, 2, 3, 4, 5, 6, 7ordtypelem2 7234 . . . . . 6  |-  ( ph  ->  Ord  T )
13 ordsson 4581 . . . . . 6  |-  ( Ord 
T  ->  T  C_  On )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  T  C_  On )
1511, 14syl5ss 3190 . . . 4  |-  ( ph  ->  ( T  i^i  dom  F )  C_  On )
1610, 15eqsstrd 3212 . . 3  |-  ( ph  ->  dom  O  C_  On )
17 epweon 4575 . . . 4  |-  _E  We  On
18 weso 4384 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
1917, 18ax-mp 8 . . 3  |-  _E  Or  On
20 soss 4332 . . 3  |-  ( dom 
O  C_  On  ->  (  _E  Or  On  ->  _E  Or  dom  O ) )
2116, 19, 20ee10 1366 . 2  |-  ( ph  ->  _E  Or  dom  O
)
22 frn 5395 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
238, 22syl 15 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
24 wess 4380 . . . 4  |-  ( ran 
O  C_  A  ->  ( R  We  A  ->  R  We  ran  O ) )
2523, 6, 24sylc 56 . . 3  |-  ( ph  ->  R  We  ran  O
)
26 weso 4384 . . 3  |-  ( R  We  ran  O  ->  R  Or  ran  O )
27 sopo 4331 . . 3  |-  ( R  Or  ran  O  ->  R  Po  ran  O )
2825, 26, 273syl 18 . 2  |-  ( ph  ->  R  Po  ran  O
)
29 ffun 5391 . . . 4  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
308, 29syl 15 . . 3  |-  ( ph  ->  Fun  O )
31 funforn 5458 . . 3  |-  ( Fun 
O  <->  O : dom  O -onto-> ran  O )
3230, 31sylib 188 . 2  |-  ( ph  ->  O : dom  O -onto-> ran  O )
33 epel 4308 . . . . 5  |-  ( a  _E  b  <->  a  e.  b )
341, 2, 3, 4, 5, 6, 7ordtypelem6 7238 . . . . 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 2626 . . 3  |-  ( ph  ->  A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
3736ralrimivw 2627 . 2  |-  ( ph  ->  A. a  e.  dom  O A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
38 soisoi 5825 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077    _E cep 4303    Po wpo 4312    Or wor 4313   Se wse 4350    We wwe 4351   Ord word 4391   Oncon0 4392   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249   -->wf 5251   -onto->wfo 5253   ` cfv 5255    Isom wiso 5256   iota_crio 6297  recscrecs 6387  OrdIsocoi 7224
This theorem is referenced by:  ordtypelem9  7241  ordtypelem10  7242  oiiso2  7246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-oi 7225
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