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Theorem ordtypelem1 7479
Description: Lemma for ordtype 7493. (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 3737 . . 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 643 . 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 7472 . . 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 2455 . 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 5135 . 2  |-  ( F  |`  T )  =  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } )
134, 10, 123eqtr4g 2492 1  |-  ( ph  ->  O  =  ( F  |`  T ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948   (/)c0 3620   ifcif 3731   class class class wbr 4204    e. cmpt 4258   Se wse 4531    We wwe 4532   Oncon0 4573   ran crn 4871    |` cres 4872   "cima 4873   iota_crio 6534  recscrecs 6624  OrdIsocoi 7470
This theorem is referenced by:  ordtypelem4  7482  ordtypelem6  7484  ordtypelem7  7485  ordtypelem9  7487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-riota 6541  df-recs 6625  df-oi 7471
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