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Theorem ordtypelem5 7253
Description: Lemma for ordtype 7263. (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 7250 . . . 4  |-  ( ph  ->  Ord  T )
91tfr1a 6426 . . . . . 6  |-  ( Fun 
F  /\  Lim  dom  F
)
109simpri 448 . . . . 5  |-  Lim  dom  F
11 limord 4467 . . . . 5  |-  ( Lim 
dom  F  ->  Ord  dom  F )
1210, 11ax-mp 8 . . . 4  |-  Ord  dom  F
13 ordin 4438 . . . 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 7252 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
16 fdm 5409 . . . . 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 4415 . . . 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 5396 . . 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 1632   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    i^i cin 3164   class class class wbr 4039    e. cmpt 4093   Se wse 4366    We wwe 4367   Ord word 4407   Oncon0 4408   Lim wlim 4409   dom cdm 4705   ran crn 4706   "cima 4708   Fun wfun 5265   -->wf 5267   iota_crio 6313  recscrecs 6403  OrdIsocoi 7240
This theorem is referenced by:  oicl  7260  oif  7261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-recs 6404  df-oi 7241
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