MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  umgraf2 Structured version   Unicode version

Theorem umgraf2 24438
Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgraf 24439 without explicitly specified domain of the edge function (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 24435 . . . 4  |-  Rel UMGrph
21brrelexi 4954 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
31brrelex2i 4955 . . 3  |-  ( V UMGrph  E  ->  E  e.  _V )
4 isumgra 24436 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
52, 3, 4syl2anc 659 . 2  |-  ( V UMGrph  E  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
65ibi 241 1  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1826   {crab 2736   _Vcvv 3034    \ cdif 3386   (/)c0 3711   ~Pcpw 3927   {csn 3944   class class class wbr 4367   dom cdm 4913   -->wf 5492   ` cfv 5496    <_ cle 9540   2c2 10502   #chash 12307   UMGrph cumg 24433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-fun 5498  df-fn 5499  df-f 5500  df-umgra 24434
This theorem is referenced by:  umgraf  24439  umgrares  24445  eupacl  25090  eupapf  25093  eupaseg  25094  eupares  25096  eupath  25102
  Copyright terms: Public domain W3C validator