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Theorem fnwe 6429
Description: A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
fnwe.1  |-  T  =  { <. x ,  y
>.  |  ( (
x  e.  A  /\  y  e.  A )  /\  ( ( F `  x ) R ( F `  y )  \/  ( ( F `
 x )  =  ( F `  y
)  /\  x S
y ) ) ) }
fnwe.2  |-  ( ph  ->  F : A --> B )
fnwe.3  |-  ( ph  ->  R  We  B )
fnwe.4  |-  ( ph  ->  S  We  A )
fnwe.5  |-  ( ph  ->  ( F " w
)  e.  _V )
Assertion
Ref Expression
fnwe  |-  ( ph  ->  T  We  A )
Distinct variable groups:    x, w, y, A    w, B, x, y    ph, w, x    w, F, x, y    w, R, x, y    w, S, x, y    w, T
Allowed substitution hints:    ph( y)    T( x, y)

Proof of Theorem fnwe
Dummy variables  u  v  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnwe.1 . 2  |-  T  =  { <. x ,  y
>.  |  ( (
x  e.  A  /\  y  e.  A )  /\  ( ( F `  x ) R ( F `  y )  \/  ( ( F `
 x )  =  ( F `  y
)  /\  x S
y ) ) ) }
2 fnwe.2 . 2  |-  ( ph  ->  F : A --> B )
3 fnwe.3 . 2  |-  ( ph  ->  R  We  B )
4 fnwe.4 . 2  |-  ( ph  ->  S  We  A )
5 fnwe.5 . 2  |-  ( ph  ->  ( F " w
)  e.  _V )
6 eqid 2412 . 2  |-  { <. u ,  v >.  |  ( ( u  e.  ( B  X.  A )  /\  v  e.  ( B  X.  A ) )  /\  ( ( 1st `  u ) R ( 1st `  v
)  \/  ( ( 1st `  u )  =  ( 1st `  v
)  /\  ( 2nd `  u ) S ( 2nd `  v ) ) ) ) }  =  { <. u ,  v >.  |  ( ( u  e.  ( B  X.  A )  /\  v  e.  ( B  X.  A ) )  /\  ( ( 1st `  u ) R ( 1st `  v
)  \/  ( ( 1st `  u )  =  ( 1st `  v
)  /\  ( 2nd `  u ) S ( 2nd `  v ) ) ) ) }
7 eqid 2412 . 2  |-  ( z  e.  A  |->  <. ( F `  z ) ,  z >. )  =  ( z  e.  A  |->  <. ( F `  z ) ,  z
>. )
81, 2, 3, 4, 5, 6, 7fnwelem 6428 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785   class class class wbr 4180   {copab 4233    e. cmpt 4234    We wwe 4508    X. cxp 4843   "cima 4848   -->wf 5417   ` cfv 5421   1stc1st 6314   2ndc2nd 6315
This theorem is referenced by:  r0weon  7858
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-int 4019  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-1st 6316  df-2nd 6317
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