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Theorem ordtypelem10 7332
Description: Lemma for ordtype 7337. Using ax-rep 4212, exclude the possibility that  O is a proper class and does not enumerate all of 
A. (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
ordtypelem10  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
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 ordtypelem10
Dummy variables  b 
c  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . 3  |-  F  = recs ( G )
2 ordtypelem.2 . . 3  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 ordtypelem.3 . . 3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )
4 ordtypelem.5 . . 3  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
5 ordtypelem.6 . . 3  |-  O  = OrdIso
( R ,  A
)
6 ordtypelem.7 . . 3  |-  ( ph  ->  R  We  A )
7 ordtypelem.8 . . 3  |-  ( ph  ->  R Se  A )
81, 2, 3, 4, 5, 6, 7ordtypelem8 7330 . 2  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
91, 2, 3, 4, 5, 6, 7ordtypelem4 7326 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
10 frn 5478 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
119, 10syl 15 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
12 simprl 732 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  A
)
136adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R  We  A
)
147adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R Se  A )
151, 2, 3, 4, 5, 13, 14ordtypelem8 7330 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
16 isof1o 5909 . . . . . . . . . . . . . 14  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  ran  O )  ->  O : dom  O -1-1-onto-> ran  O )
17 f1of 5555 . . . . . . . . . . . . . 14  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O --> ran  O )
1815, 16, 173syl 18 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O --> ran  O )
19 f1of1 5554 . . . . . . . . . . . . . . 15  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O -1-1-> ran 
O )
2015, 16, 193syl 18 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O
-1-1-> ran  O )
2111adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  A
)
22 rexnal 2630 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. m  e.  dom  O  -.  ( O `  m
) R b  <->  -.  A. m  e.  dom  O ( O `
 m ) R b )
231, 2, 3, 4, 5, 6, 7ordtypelem7 7329 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( ( O `  m ) R b  \/  b  e.  ran  O ) )
2423ord 366 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( -.  ( O `
 m ) R b  ->  b  e.  ran  O ) )
2524rexlimdva 2743 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  b  e.  A )  ->  ( E. m  e.  dom  O  -.  ( O `  m ) R b  ->  b  e.  ran  O ) )
2622, 25syl5bir 209 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  A. m  e.  dom  O ( O `  m
) R b  -> 
b  e.  ran  O
) )
2726con1d 116 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  A. m  e.  dom  O ( O `  m
) R b ) )
2827impr 602 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. m  e.  dom  O ( O `  m
) R b )
29 ffun 5474 . . . . . . . . . . . . . . . . . . . . 21  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
309, 29syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  Fun  O )
31 funfn 5365 . . . . . . . . . . . . . . . . . . . 20  |-  ( Fun 
O  <->  O  Fn  dom  O )
3230, 31sylib 188 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  O  Fn  dom  O
)
3332adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Fn  dom  O )
34 breq1 4107 . . . . . . . . . . . . . . . . . . 19  |-  ( c  =  ( O `  m )  ->  (
c R b  <->  ( O `  m ) R b ) )
3534ralrn 5751 . . . . . . . . . . . . . . . . . 18  |-  ( O  Fn  dom  O  -> 
( A. c  e. 
ran  O  c R
b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
3633, 35syl 15 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ( A. c  e.  ran  O  c R b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
3728, 36mpbird 223 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. c  e.  ran  O  c R b )
38 ssrab 3327 . . . . . . . . . . . . . . . 16  |-  ( ran 
O  C_  { c  e.  A  |  c R b }  <->  ( ran  O 
C_  A  /\  A. c  e.  ran  O  c R b ) )
3921, 37, 38sylanbrc 645 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  { c  e.  A  |  c R b } )
40 simpl 443 . . . . . . . . . . . . . . . 16  |-  ( ( b  e.  A  /\  -.  b  e.  ran  O )  ->  b  e.  A )
41 seex 4438 . . . . . . . . . . . . . . . 16  |-  ( ( R Se  A  /\  b  e.  A )  ->  { c  e.  A  |  c R b }  e.  _V )
427, 40, 41syl2an 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  { c  e.  A  |  c R b }  e.  _V )
43 ssexg 4241 . . . . . . . . . . . . . . 15  |-  ( ( ran  O  C_  { c  e.  A  |  c R b }  /\  { c  e.  A  | 
c R b }  e.  _V )  ->  ran  O  e.  _V )
4439, 42, 43syl2anc 642 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  e.  _V )
45 f1dmex 5837 . . . . . . . . . . . . . 14  |-  ( ( O : dom  O -1-1-> ran 
O  /\  ran  O  e. 
_V )  ->  dom  O  e.  _V )
4620, 44, 45syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  dom  O  e.  _V )
47 fex 5835 . . . . . . . . . . . . 13  |-  ( ( O : dom  O --> ran  O  /\  dom  O  e.  _V )  ->  O  e.  _V )
4818, 46, 47syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  e.  _V )
491, 2, 3, 4, 5, 13, 14, 48ordtypelem9 7331 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  A ) )
50 isof1o 5909 . . . . . . . . . . 11  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  A
)  ->  O : dom  O -1-1-onto-> A )
5149, 50syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O -1-1-onto-> A )
52 f1ofo 5562 . . . . . . . . . 10  |-  ( O : dom  O -1-1-onto-> A  ->  O : dom  O -onto-> A
)
53 forn 5537 . . . . . . . . . 10  |-  ( O : dom  O -onto-> A  ->  ran  O  =  A )
5451, 52, 533syl 18 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  =  A )
5512, 54eleqtrrd 2435 . . . . . . . 8  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  ran  O )
5655expr 598 . . . . . . 7  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  b  e.  ran  O ) )
5756pm2.18d 103 . . . . . 6  |-  ( (
ph  /\  b  e.  A )  ->  b  e.  ran  O )
5857ex 423 . . . . 5  |-  ( ph  ->  ( b  e.  A  ->  b  e.  ran  O
) )
5958ssrdv 3261 . . . 4  |-  ( ph  ->  A  C_  ran  O )
6011, 59eqssd 3272 . . 3  |-  ( ph  ->  ran  O  =  A )
61 isoeq5 5907 . . 3  |-  ( ran 
O  =  A  -> 
( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
6260, 61syl 15 . 2  |-  ( ph  ->  ( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
638, 62mpbid 201 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ 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   Se wse 4432    We wwe 4433   Oncon0 4474   dom cdm 4771   ran crn 4772   "cima 4774   Fun wfun 5331    Fn wfn 5332   -->wf 5333   -1-1->wf1 5334   -onto->wfo 5335   -1-1-onto->wf1o 5336   ` cfv 5337    Isom wiso 5338   iota_crio 6384  recscrecs 6474  OrdIsocoi 7314
This theorem is referenced by:  ordtype  7337
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-rep 4212  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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