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Theorem ordtypelem5 7492
Description: Lemma for ordtype 7502. (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 7489 . . . 4  |-  ( ph  ->  Ord  T )
91tfr1a 6656 . . . . . 6  |-  ( Fun 
F  /\  Lim  dom  F
)
109simpri 450 . . . . 5  |-  Lim  dom  F
11 limord 4641 . . . . 5  |-  ( Lim 
dom  F  ->  Ord  dom  F )
1210, 11ax-mp 8 . . . 4  |-  Ord  dom  F
13 ordin 4612 . . . 4  |-  ( ( Ord  T  /\  Ord  dom 
F )  ->  Ord  ( T  i^i  dom  F
) )
148, 12, 13sylancl 645 . . 3  |-  ( ph  ->  Ord  ( T  i^i  dom 
F ) )
151, 2, 3, 4, 5, 6, 7ordtypelem4 7491 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
16 fdm 5596 . . . . 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 4589 . . . 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 225 . 2  |-  ( ph  ->  Ord  dom  O )
2117feq2d 5582 . . 3  |-  ( ph  ->  ( O : dom  O --> A  <->  O : ( T  i^i  dom  F ) --> A ) )
2215, 21mpbird 225 . 2  |-  ( ph  ->  O : dom  O --> A )
2320, 22jca 520 1  |-  ( ph  ->  ( Ord  dom  O  /\  O : dom  O --> A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653   A.wral 2706   E.wrex 2707   {crab 2710   _Vcvv 2957    i^i cin 3320   class class class wbr 4213    e. cmpt 4267   Se wse 4540    We wwe 4541   Ord word 4581   Oncon0 4582   Lim wlim 4583   dom cdm 4879   ran crn 4880   "cima 4882   Fun wfun 5449   -->wf 5451   iota_crio 6543  recscrecs 6633  OrdIsocoi 7479
This theorem is referenced by:  oicl  7499  oif  7500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-riota 6550  df-recs 6634  df-oi 7480
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