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Theorem ordtypelem10 7499
Description: Lemma for ordtype 7504. Using ax-rep 4323, 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 7497 . 2  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
91, 2, 3, 4, 5, 6, 7ordtypelem4 7493 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
10 frn 5600 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
119, 10syl 16 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
12 simprl 734 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  A
)
136adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R  We  A
)
147adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R Se  A )
151, 2, 3, 4, 5, 13, 14ordtypelem8 7497 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
16 isof1o 6048 . . . . . . . . . . . . 13  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  ran  O )  ->  O : dom  O -1-1-onto-> ran  O )
17 f1of 5677 . . . . . . . . . . . . 13  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O --> ran  O )
1815, 16, 173syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O --> ran  O )
19 f1of1 5676 . . . . . . . . . . . . . 14  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O -1-1-> ran 
O )
2015, 16, 193syl 19 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O
-1-1-> ran  O )
21 simpl 445 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  A  /\  -.  b  e.  ran  O )  ->  b  e.  A )
22 seex 4548 . . . . . . . . . . . . . . 15  |-  ( ( R Se  A  /\  b  e.  A )  ->  { c  e.  A  |  c R b }  e.  _V )
237, 21, 22syl2an 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  { c  e.  A  |  c R b }  e.  _V )
2411adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  A
)
25 rexnal 2718 . . . . . . . . . . . . . . . . . . 19  |-  ( E. m  e.  dom  O  -.  ( O `  m
) R b  <->  -.  A. m  e.  dom  O ( O `
 m ) R b )
261, 2, 3, 4, 5, 6, 7ordtypelem7 7496 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( ( O `  m ) R b  \/  b  e.  ran  O ) )
2726ord 368 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( -.  ( O `
 m ) R b  ->  b  e.  ran  O ) )
2827rexlimdva 2832 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  b  e.  A )  ->  ( E. m  e.  dom  O  -.  ( O `  m ) R b  ->  b  e.  ran  O ) )
2925, 28syl5bir 211 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  A. m  e.  dom  O ( O `  m
) R b  -> 
b  e.  ran  O
) )
3029con1d 119 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  A. m  e.  dom  O ( O `  m
) R b ) )
3130impr 604 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. m  e.  dom  O ( O `  m
) R b )
32 ffun 5596 . . . . . . . . . . . . . . . . . . . 20  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
339, 32syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  Fun  O )
34 funfn 5485 . . . . . . . . . . . . . . . . . . 19  |-  ( Fun 
O  <->  O  Fn  dom  O )
3533, 34sylib 190 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  O  Fn  dom  O
)
3635adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Fn  dom  O )
37 breq1 4218 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  ( O `  m )  ->  (
c R b  <->  ( O `  m ) R b ) )
3837ralrn 5876 . . . . . . . . . . . . . . . . 17  |-  ( O  Fn  dom  O  -> 
( A. c  e. 
ran  O  c R
b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
3936, 38syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
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 ) )
4031, 39mpbird 225 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. c  e.  ran  O  c R b )
41 ssrab 3423 . . . . . . . . . . . . . . 15  |-  ( ran 
O  C_  { c  e.  A  |  c R b }  <->  ( ran  O 
C_  A  /\  A. c  e.  ran  O  c R b ) )
4224, 40, 41sylanbrc 647 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  { c  e.  A  |  c R b } )
4323, 42ssexd 4353 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  e.  _V )
44 f1dmex 5974 . . . . . . . . . . . . 13  |-  ( ( O : dom  O -1-1-> ran 
O  /\  ran  O  e. 
_V )  ->  dom  O  e.  _V )
4520, 43, 44syl2anc 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  dom  O  e.  _V )
46 fex 5972 . . . . . . . . . . . 12  |-  ( ( O : dom  O --> ran  O  /\  dom  O  e.  _V )  ->  O  e.  _V )
4718, 45, 46syl2anc 644 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  e.  _V )
481, 2, 3, 4, 5, 13, 14, 47ordtypelem9 7498 . . . . . . . . . 10  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  A ) )
49 isof1o 6048 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  A
)  ->  O : dom  O -1-1-onto-> A )
50 f1ofo 5684 . . . . . . . . . 10  |-  ( O : dom  O -1-1-onto-> A  ->  O : dom  O -onto-> A
)
51 forn 5659 . . . . . . . . . 10  |-  ( O : dom  O -onto-> A  ->  ran  O  =  A )
5248, 49, 50, 514syl 20 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  =  A )
5312, 52eleqtrrd 2515 . . . . . . . 8  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  ran  O )
5453expr 600 . . . . . . 7  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  b  e.  ran  O ) )
5554pm2.18d 106 . . . . . 6  |-  ( (
ph  /\  b  e.  A )  ->  b  e.  ran  O )
5655ex 425 . . . . 5  |-  ( ph  ->  ( b  e.  A  ->  b  e.  ran  O
) )
5756ssrdv 3356 . . . 4  |-  ( ph  ->  A  C_  ran  O )
5811, 57eqssd 3367 . . 3  |-  ( ph  ->  ran  O  =  A )
59 isoeq5 6046 . . 3  |-  ( ran 
O  =  A  -> 
( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
6058, 59syl 16 . 2  |-  ( ph  ->  ( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
618, 60mpbid 203 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    i^i cin 3321    C_ wss 3322   class class class wbr 4215    e. cmpt 4269    _E cep 4495   Se wse 4542    We wwe 4543   Oncon0 4584   dom cdm 4881   ran crn 4882   "cima 4884   Fun wfun 5451    Fn wfn 5452   -->wf 5453   -1-1->wf1 5454   -onto->wfo 5455   -1-1-onto->wf1o 5456   ` cfv 5457    Isom wiso 5458   iota_crio 6545  recscrecs 6635  OrdIsocoi 7481
This theorem is referenced by:  ordtype  7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-oi 7482
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