MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnwe Structured version   Unicode version

Theorem fnwe 6889
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 2454 . 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 2454 . 2  |-  ( z  e.  A  |->  <. ( F `  z ) ,  z >. )  =  ( z  e.  A  |->  <. ( F `  z ) ,  z
>. )
81, 2, 3, 4, 5, 6, 7fnwelem 6888 1  |-  ( ph  ->  T  We  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022   class class class wbr 4439   {copab 4496    |-> cmpt 4497    We wwe 4826    X. cxp 4986   "cima 4991   -->wf 5566   ` cfv 5570   1stc1st 6771   2ndc2nd 6772
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-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-1st 6773  df-2nd 6774
This theorem is referenced by:  r0weon  8381
  Copyright terms: Public domain W3C validator