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Theorem releupa 23880
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 23864 . 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 6201 1  |-  Rel  ( V EulPaths  E )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    = wceq 1623   A.wral 2543   E.wrex 2544   _Vcvv 2788   {cpr 3641   class class class wbr 4023   dom cdm 4689   Rel wrel 4694   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    - cmin 9037   NN0cn0 9965   ...cfz 10782   UMGrph cumg 23860   EulPaths ceup 23861
This theorem is referenced by:  iseupa  23881  eupath  23905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-eupa 23864
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