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Theorem umgraf2 23884
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 23881 . . . 4  |-  Rel UMGrph
21brrelexi 4745 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
31brrelex2i 4746 . . 3  |-  ( V UMGrph  E  ->  E  e.  _V )
4 isumgra 23882 . . 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 1696   {crab 2560   _Vcvv 2801    \ cdif 3162   (/)c0 3468   ~Pcpw 3638   {csn 3653   class class class wbr 4039   dom cdm 4705   -->wf 5267   ` cfv 5271    <_ cle 8884   2c2 9811   #chash 11353   UMGrph cumg 23875
This theorem is referenced by:  umgraf  23885  umgrares  23891  eupacl  23899  eupapf  23902  eupaseg  23903  eupares  23914  eupath  23920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-umgra 23878
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