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

Theorem releupa 25106
Description: The set  ( V EulPaths  E ) of all Eulerian paths on  <. V ,  E >. is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
releupa  |-  Rel  ( V EulPaths  E )

Proof of Theorem releupa
Dummy variables  e 
f  k  n  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-eupa 25105 . 2  |- EulPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( v UMGrph  e  /\  E. n  e.  NN0  (
f : ( 1 ... n ) -1-1-onto-> dom  e  /\  p : ( 0 ... n ) --> v  /\  A. k  e.  ( 1 ... n
) ( e `  ( f `  k
) )  =  {
( p `  (
k  -  1 ) ) ,  ( p `
 k ) } ) ) } )
21relmpt2opab 6799 1  |-  Rel  ( V EulPaths  E )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    /\ w3a 971    = wceq 1399   A.wral 2742   E.wrex 2743   _Vcvv 3047   {cpr 3959   class class class wbr 4380   dom cdm 4926   Rel wrel 4931   -->wf 5505   -1-1-onto->wf1o 5508   ` cfv 5509  (class class class)co 6214   0cc0 9421   1c1 9422    - cmin 9736   NN0cn0 10730   ...cfz 11611   UMGrph cumg 24454   EulPaths ceup 25104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fv 5517  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-1st 6717  df-2nd 6718  df-eupa 25105
This theorem is referenced by:  eupath  25123
  Copyright terms: Public domain W3C validator