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Theorem ordtypelem1 7421
Description: Lemma for ordtype 7435. (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 3689 . . 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 7414 . . 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 2408 . 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 5084 . 2  |-  ( F  |`  T )  =  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } )
134, 10, 123eqtr4g 2445 1  |-  ( ph  ->  O  =  ( F  |`  T ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649   A.wral 2650   E.wrex 2651   {crab 2654   _Vcvv 2900   (/)c0 3572   ifcif 3683   class class class wbr 4154    e. cmpt 4208   Se wse 4481    We wwe 4482   Oncon0 4523   ran crn 4820    |` cres 4821   "cima 4822   iota_crio 6479  recscrecs 6569  OrdIsocoi 7412
This theorem is referenced by:  ordtypelem4  7424  ordtypelem6  7426  ordtypelem7  7427  ordtypelem9  7429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-xp 4825  df-cnv 4827  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fv 5403  df-riota 6486  df-recs 6570  df-oi 7413
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