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Theorem ordtypelem5 7735
Description: Lemma for ordtype 7745. (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
ordtypelem5  |-  ( ph  ->  ( Ord  dom  O  /\  O : 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 ordtypelem5
StepHypRef Expression
1 ordtypelem.1 . . . . 5  |-  F  = recs ( G )
2 ordtypelem.2 . . . . 5  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 ordtypelem.3 . . . . 5  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
4 ordtypelem.5 . . . . 5  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
5 ordtypelem.6 . . . . 5  |-  O  = OrdIso
( R ,  A
)
6 ordtypelem.7 . . . . 5  |-  ( ph  ->  R  We  A )
7 ordtypelem.8 . . . . 5  |-  ( ph  ->  R Se  A )
81, 2, 3, 4, 5, 6, 7ordtypelem2 7732 . . . 4  |-  ( ph  ->  Ord  T )
91tfr1a 6852 . . . . . 6  |-  ( Fun 
F  /\  Lim  dom  F
)
109simpri 462 . . . . 5  |-  Lim  dom  F
11 limord 4777 . . . . 5  |-  ( Lim 
dom  F  ->  Ord  dom  F )
1210, 11ax-mp 5 . . . 4  |-  Ord  dom  F
13 ordin 4748 . . . 4  |-  ( ( Ord  T  /\  Ord  dom 
F )  ->  Ord  ( T  i^i  dom  F
) )
148, 12, 13sylancl 662 . . 3  |-  ( ph  ->  Ord  ( T  i^i  dom 
F ) )
151, 2, 3, 4, 5, 6, 7ordtypelem4 7734 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
16 fdm 5562 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  dom  O  =  ( T  i^i  dom  F )
)
1715, 16syl 16 . . . 4  |-  ( ph  ->  dom  O  =  ( T  i^i  dom  F
) )
18 ordeq 4725 . . . 4  |-  ( dom 
O  =  ( T  i^i  dom  F )  ->  ( Ord  dom  O  <->  Ord  ( T  i^i  dom  F ) ) )
1917, 18syl 16 . . 3  |-  ( ph  ->  ( Ord  dom  O  <->  Ord  ( T  i^i  dom  F ) ) )
2014, 19mpbird 232 . 2  |-  ( ph  ->  Ord  dom  O )
2117feq2d 5546 . . 3  |-  ( ph  ->  ( O : dom  O --> A  <->  O : ( T  i^i  dom  F ) --> A ) )
2215, 21mpbird 232 . 2  |-  ( ph  ->  O : dom  O --> A )
2320, 22jca 532 1  |-  ( ph  ->  ( Ord  dom  O  /\  O : dom  O --> A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   A.wral 2714   E.wrex 2715   {crab 2718   _Vcvv 2971    i^i cin 3326   class class class wbr 4291    e. cmpt 4349   Se wse 4676    We wwe 4677   Ord word 4717   Oncon0 4718   Lim wlim 4719   dom cdm 4839   ran crn 4840   "cima 4842   Fun wfun 5411   -->wf 5413   iota_crio 6050  recscrecs 6830  OrdIsocoi 7722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-recs 6831  df-oi 7723
This theorem is referenced by:  oicl  7742  oif  7743
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