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Theorem ordtypelem1 7233
Description: Lemma for ordtype 7247. (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 3571 . . 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 7226 . . 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 2303 . 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 4952 . 2  |-  ( F  |`  T )  =  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } )
134, 10, 123eqtr4g 2340 1  |-  ( ph  ->  O  =  ( F  |`  T ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788   (/)c0 3455   ifcif 3565   class class class wbr 4023    e. cmpt 4077   Se wse 4350    We wwe 4351   Oncon0 4392   ran crn 4690    |` cres 4691   "cima 4692   iota_crio 6297  recscrecs 6387  OrdIsocoi 7224
This theorem is referenced by:  ordtypelem4  7236  ordtypelem6  7238  ordtypelem7  7239  ordtypelem9  7241
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-riota 6304  df-recs 6388  df-oi 7225
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