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Theorem isumgra 21217
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 448 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  e  =  E )
21dmeqd 5012 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  dom  e  =  dom  E )
31, 2feq12d 5522 . . 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 444 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
54pweqd 3747 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
65difeq1d 3407 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ~P v  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
7 rabeq 2893 . . . 4  |-  ( ( ~P v  \  { (/)
} )  =  ( ~P V  \  { (/)
} )  ->  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }  =  { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
8 feq3 5518 . . . 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 19 . . 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 245 . 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 21215 . 2  |- UMGrph  =  { <. v ,  e >.  |  e : dom  e
--> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
1210, 11brabga 4410 1  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2653    \ cdif 3260   (/)c0 3571   ~Pcpw 3742   {csn 3757   class class class wbr 4153   dom cdm 4818   -->wf 5390   ` cfv 5394    <_ cle 9054   2c2 9981   #chash 11545   UMGrph cumg 21214
This theorem is referenced by:  wrdumgra  21218  umgraf2  21219  umgrares  21226  umgra0  21227  umgra1  21228  umisuhgra  21229  umgraun  21230  uslisumgra  21253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-fun 5396  df-fn 5397  df-f 5398  df-umgra 21215
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