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Theorem fnwe 6247
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 2296 . 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 2296 . 2  |-  ( z  e.  A  |->  <. ( F `  z ) ,  z >. )  =  ( z  e.  A  |->  <. ( F `  z ) ,  z
>. )
81, 2, 3, 4, 5, 6, 7fnwelem 6246 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   {copab 4092    e. cmpt 4093    We wwe 4367    X. cxp 4703   "cima 4708   -->wf 5267   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  r0weon  7656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-1st 6138  df-2nd 6139
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