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Theorem fnwe 6231
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 2283 . 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 2283 . 2  |-  ( z  e.  A  |->  <. ( F `  z ) ,  z >. )  =  ( z  e.  A  |->  <. ( F `  z ) ,  z
>. )
81, 2, 3, 4, 5, 6, 7fnwelem 6230 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   {copab 4076    e. cmpt 4077    We wwe 4351    X. cxp 4687   "cima 4692   -->wf 5251   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  r0weon  7640
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-1st 6122  df-2nd 6123
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