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Theorem isumgra 23402
Description: The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
isumgra  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
Distinct variable groups:    x, E    x, V    x, W
Allowed substitution hint:    X( x)

Proof of Theorem isumgra
Dummy variables  v 
e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  e  =  E )
21dmeqd 5151 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
31, 2feq12d 5657 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e : dom  e
--> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 }  <->  E : dom  E --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
4 simpl 457 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
54pweqd 3974 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
65difeq1d 3582 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
7 rabeq 3072 . . . 4  |-  ( ( ~P v  \  { (/)
} )  =  ( ~P V  \  { (/)
} )  ->  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }  =  { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
8 feq3 5653 . . . 4  |-  ( { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 }  =  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  ( E : dom  E --> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 }  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
96, 7, 83syl 20 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( E : dom  E --> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 }  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
103, 9bitrd 253 . 2  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e : dom  e
--> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 }  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
11 df-umgra 23400 . 2  |- UMGrph  =  { <. v ,  e >.  |  e : dom  e
--> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
1210, 11brabga 4712 1  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2803    \ cdif 3434   (/)c0 3746   ~Pcpw 3969   {csn 3986   class class class wbr 4401   dom cdm 4949   -->wf 5523   ` cfv 5527    <_ cle 9531   2c2 10483   #chash 12221   UMGrph cumg 23399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-fun 5529  df-fn 5530  df-f 5531  df-umgra 23400
This theorem is referenced by:  wrdumgra  23403  umgraf2  23404  umgrares  23411  umgra0  23412  umgra1  23413  umisuhgra  23414  umgraun  23415  uslisumgra  23438
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