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Theorem ordtypelem1 7935
Description: Lemma for ordtype 7949. (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 3935 . . 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 659 . 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 7928 . . 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 2483 . 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 5259 . 2  |-  ( F  |`  T )  =  ( F  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t } )
134, 10, 123eqtr4g 2520 1  |-  ( ph  ->  O  =  ( F  |`  T ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   (/)c0 3783   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   Se wse 4825    We wwe 4826   Oncon0 4867   ran crn 4989    |` cres 4990   "cima 4991   iota_crio 6231  recscrecs 7033  OrdIsocoi 7926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578  df-riota 6232  df-recs 7034  df-oi 7927
This theorem is referenced by:  ordtypelem4  7938  ordtypelem6  7940  ordtypelem7  7941  ordtypelem9  7943
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