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Theorem fnwe 6793
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 2452 . 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 2452 . 2  |-  ( z  e.  A  |->  <. ( F `  z ) ,  z >. )  =  ( z  e.  A  |->  <. ( F `  z ) ,  z
>. )
81, 2, 3, 4, 5, 6, 7fnwelem 6792 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3072   <.cop 3986   class class class wbr 4395   {copab 4452    |-> cmpt 4453    We wwe 4781    X. cxp 4941   "cima 4946   -->wf 5517   ` cfv 5521   1stc1st 6680   2ndc2nd 6681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-int 4232  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-1st 6682  df-2nd 6683
This theorem is referenced by:  r0weon  8285
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