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Theorem ordtypelem5 7862
Description: Lemma for ordtype 7872. (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 7859 . . . 4  |-  ( ph  ->  Ord  T )
91tfr1a 6981 . . . . . 6  |-  ( Fun 
F  /\  Lim  dom  F
)
109simpri 460 . . . . 5  |-  Lim  dom  F
11 limord 4851 . . . . 5  |-  ( Lim 
dom  F  ->  Ord  dom  F )
1210, 11ax-mp 5 . . . 4  |-  Ord  dom  F
13 ordin 4822 . . . 4  |-  ( ( Ord  T  /\  Ord  dom 
F )  ->  Ord  ( T  i^i  dom  F
) )
148, 12, 13sylancl 660 . . 3  |-  ( ph  ->  Ord  ( T  i^i  dom 
F ) )
151, 2, 3, 4, 5, 6, 7ordtypelem4 7861 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
16 fdm 5643 . . . . 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 4799 . . . 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 5626 . . 3  |-  ( ph  ->  ( O : dom  O --> A  <->  O : ( T  i^i  dom  F ) --> A ) )
2215, 21mpbird 232 . 2  |-  ( ph  ->  O : dom  O --> A )
2320, 22jca 530 1  |-  ( ph  ->  ( Ord  dom  O  /\  O : dom  O --> A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399   A.wral 2732   E.wrex 2733   {crab 2736   _Vcvv 3034    i^i cin 3388   class class class wbr 4367    |-> cmpt 4425   Se wse 4750    We wwe 4751   Ord word 4791   Oncon0 4792   Lim wlim 4793   dom cdm 4913   ran crn 4914   "cima 4916   Fun wfun 5490   -->wf 5492   iota_crio 6157  recscrecs 6959  OrdIsocoi 7849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-recs 6960  df-oi 7850
This theorem is referenced by:  oicl  7869  oif  7870
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