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Theorem umgraf2 21352
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 21349 . . . 4  |-  Rel UMGrph
21brrelexi 4918 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
31brrelex2i 4919 . . 3  |-  ( V UMGrph  E  ->  E  e.  _V )
4 isumgra 21350 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
52, 3, 4syl2anc 643 . 2  |-  ( V UMGrph  E  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
65ibi 233 1  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   {crab 2709   _Vcvv 2956    \ cdif 3317   (/)c0 3628   ~Pcpw 3799   {csn 3814   class class class wbr 4212   dom cdm 4878   -->wf 5450   ` cfv 5454    <_ cle 9121   2c2 10049   #chash 11618   UMGrph cumg 21347
This theorem is referenced by:  umgraf  21353  umgrares  21359  eupacl  21691  eupapf  21694  eupaseg  21695  eupares  21697  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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-f 5458  df-umgra 21348
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