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Theorem releupa 21686
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 21685 . 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 6429 1  |-  Rel  ( V EulPaths  E )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    = wceq 1652   A.wral 2705   E.wrex 2706   _Vcvv 2956   {cpr 3815   class class class wbr 4212   dom cdm 4878   Rel wrel 4883   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    - cmin 9291   NN0cn0 10221   ...cfz 11043   UMGrph cumg 21347   EulPaths ceup 21684
This theorem is referenced by:  eupath  21703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-eupa 21685
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