Theorem fnwe 6916

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
fnwe.2
fnwe.3
fnwe.4
fnwe.5

Assertion

Ref	Expression
fnwe

Distinct variable groups:   ,,,   ,,,   ,,   ,,,   ,,,   ,S,,   ,

Proof of Theorem fnwe

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnwe.1	. 2
2		fnwe.2	. 2
3		fnwe.3	. 2
4		fnwe.4	. 2
5		fnwe.5	. 2
6		eqid 2457	. 2
7		eqid 2457	. 2
8	1, 2, 3, 4, 5, 6, 7	fnwelem 6915	1

Syntax hints:  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  cvv 3109  <.cop 4035   class class class wbr 4452  {copab 4509  e.cmpt 4510  Wewwe 4842  X.cxp 5002  "cima 5007  -->wf 5589  `cfv 5593  c1st 6798  c2nd 6799

This theorem is referenced by:  r0weon  8411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-int 4287  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-1st 6800  df-2nd 6801