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Theorem umgraf2 23869
Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
umgraf2  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
Distinct variable groups:    x, E    x, V

Proof of Theorem umgraf2
StepHypRef Expression
1 relumgra 23866 . . . 4  |-  Rel UMGrph
21brrelexi 4729 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
31brrelex2i 4730 . . 3  |-  ( V UMGrph  E  ->  E  e.  _V )
4 isumgra 23867 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
52, 3, 4syl2anc 642 . 2  |-  ( V UMGrph  E  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
65ibi 232 1  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023   dom cdm 4689   -->wf 5251   ` cfv 5255    <_ cle 8868   2c2 9795   #chash 11337   UMGrph cumg 23860
This theorem is referenced by:  umgraf  23870  umgrares  23876  eupacl  23884  eupapf  23887  eupaseg  23888  eupares  23899  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-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-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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-umgra 23863
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