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Theorem ordtypelem5 7237
Description: Lemma for ordtype 7247. (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 7234 . . . 4  |-  ( ph  ->  Ord  T )
91tfr1a 6410 . . . . . 6  |-  ( Fun 
F  /\  Lim  dom  F
)
109simpri 448 . . . . 5  |-  Lim  dom  F
11 limord 4451 . . . . 5  |-  ( Lim 
dom  F  ->  Ord  dom  F )
1210, 11ax-mp 8 . . . 4  |-  Ord  dom  F
13 ordin 4422 . . . 4  |-  ( ( Ord  T  /\  Ord  dom 
F )  ->  Ord  ( T  i^i  dom  F
) )
148, 12, 13sylancl 643 . . 3  |-  ( ph  ->  Ord  ( T  i^i  dom 
F ) )
151, 2, 3, 4, 5, 6, 7ordtypelem4 7236 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
16 fdm 5393 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  dom  O  =  ( T  i^i  dom  F )
)
1715, 16syl 15 . . . 4  |-  ( ph  ->  dom  O  =  ( T  i^i  dom  F
) )
18 ordeq 4399 . . . 4  |-  ( dom 
O  =  ( T  i^i  dom  F )  ->  ( Ord  dom  O  <->  Ord  ( T  i^i  dom  F ) ) )
1917, 18syl 15 . . 3  |-  ( ph  ->  ( Ord  dom  O  <->  Ord  ( T  i^i  dom  F ) ) )
2014, 19mpbird 223 . 2  |-  ( ph  ->  Ord  dom  O )
2117feq2d 5380 . . 3  |-  ( ph  ->  ( O : dom  O --> A  <->  O : ( T  i^i  dom  F ) --> A ) )
2215, 21mpbird 223 . 2  |-  ( ph  ->  O : dom  O --> A )
2320, 22jca 518 1  |-  ( ph  ->  ( Ord  dom  O  /\  O : dom  O --> A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    i^i cin 3151   class class class wbr 4023    e. cmpt 4077   Se wse 4350    We wwe 4351   Ord word 4391   Oncon0 4392   Lim wlim 4393   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249   -->wf 5251   iota_crio 6297  recscrecs 6387  OrdIsocoi 7224
This theorem is referenced by:  oicl  7244  oif  7245
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-riota 6304  df-recs 6388  df-oi 7225
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