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Theorem ordtypelem1 7249
Description: Lemma for ordtype 7263. (Contributed by Mario Carneiro, 24-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
ordtypelem1  |-  ( ph  ->  O  =  ( F  |`  T ) )
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 ordtypelem1
StepHypRef Expression
1 ordtypelem.7 . . 3  |-  ( ph  ->  R  We  A )
2 ordtypelem.8 . . 3  |-  ( ph  ->  R Se  A )
3 iftrue 3584 . . 3  |-  ( ( R  We  A  /\  R Se  A )  ->  if ( ( R  We  A  /\  R Se  A ) ,  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t } ) ,  (/) )  =  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } ) )
41, 2, 3syl2anc 642 . 2  |-  ( ph  ->  if ( ( R  We  A  /\  R Se  A ) ,  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } ) ,  (/) )  =  ( F  |` 
{ x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t } ) )
5 ordtypelem.6 . . 3  |-  O  = OrdIso
( R ,  A
)
6 ordtypelem.2 . . . 4  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
7 ordtypelem.3 . . . 4  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )
8 ordtypelem.1 . . . 4  |-  F  = recs ( G )
96, 7, 8dfoi 7242 . . 3  |- OrdIso ( R ,  A )  =  if ( ( R  We  A  /\  R Se  A ) ,  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } ) ,  (/) )
105, 9eqtri 2316 . 2  |-  O  =  if ( ( R  We  A  /\  R Se  A ) ,  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } ) ,  (/) )
11 ordtypelem.5 . . 3  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
1211reseq2i 4968 . 2  |-  ( F  |`  T )  =  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } )
134, 10, 123eqtr4g 2353 1  |-  ( ph  ->  O  =  ( F  |`  T ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   (/)c0 3468   ifcif 3578   class class class wbr 4039    e. cmpt 4093   Se wse 4366    We wwe 4367   Oncon0 4408   ran crn 4706    |` cres 4707   "cima 4708   iota_crio 6313  recscrecs 6403  OrdIsocoi 7240
This theorem is referenced by:  ordtypelem4  7252  ordtypelem6  7254  ordtypelem7  7255  ordtypelem9  7257
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-riota 6320  df-recs 6404  df-oi 7241
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