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Theorem relumgra 23866
Description: The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
relumgra  |-  Rel UMGrph

Proof of Theorem relumgra
Dummy variables  e 
v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-umgra 23863 . 2  |- UMGrph  =  { <. v ,  e >.  |  e : dom  e
--> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
21relopabi 4811 1  |-  Rel UMGrph
Colors of variables: wff set class
Syntax hints:   {crab 2547    \ cdif 3149   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023   dom cdm 4689   Rel wrel 4694   -->wf 5251   ` cfv 5255    <_ cle 8868   2c2 9795   #chash 11337   UMGrph cumg 23860
This theorem is referenced by:  umgraf2  23869  umgrares  23876  umgraun  23879  iseupa  23881  vdgrun  23893  eupap1  23900  eupath2lem3  23903  eupath2  23904
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-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-opab 4078  df-xp 4695  df-rel 4696  df-umgra 23863
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