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Theorem relusgra 25062
Description: The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
relusgra  |-  Rel USGrph

Proof of Theorem relusgra
Dummy variables  v 
e  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-usgra 25060 . 2  |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } }
21relopabi 4959 1  |-  Rel USGrph
Colors of variables: wff setvar class
Syntax hints:    = wceq 1444   {crab 2741    \ cdif 3401   (/)c0 3731   ~Pcpw 3951   {csn 3968   dom cdm 4834   Rel wrel 4839   -1-1->wf1 5579   ` cfv 5582   2c2 10659   #chash 12515   USGrph cusg 25057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-opab 4462  df-xp 4840  df-rel 4841  df-usgra 25060
This theorem is referenced by:  usgrav  25065  elusuhgra  25095  usgrafis  25143  usgfis  39811  usgfisALT  39815
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