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Theorem isumgra 24461
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 459 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  e  =  E )
21dmeqd 5135 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
31, 2feq12d 5645 . . 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 455 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
54pweqd 3949 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
65difeq1d 3552 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
7 rabeq 3045 . . . 4  |-  ( ( ~P v  \  { (/)
} )  =  ( ~P V  \  { (/)
} )  ->  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }  =  { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
8 feq3 5640 . . . 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 24459 . 2  |- UMGrph  =  { <. v ,  e >.  |  e : dom  e
--> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
1210, 11brabga 4692 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 367    = wceq 1399    e. wcel 1836   {crab 2750    \ cdif 3403   (/)c0 3728   ~Pcpw 3944   {csn 3961   class class class wbr 4384   dom cdm 4930   -->wf 5509   ` cfv 5513    <_ cle 9562   2c2 10524   #chash 12330   UMGrph cumg 24458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-br 4385  df-opab 4443  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-fun 5515  df-fn 5516  df-f 5517  df-umgra 24459
This theorem is referenced by:  wrdumgra  24462  umgraf2  24463  umgrares  24470  umgra0  24471  umgra1  24472  umisuhgra  24473  umgraun  24474  uslisumgra  24510
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