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Theorem umgraf 23994
Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
umgraf  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
Distinct variable groups:    x, A    x, E    x, V

Proof of Theorem umgraf
StepHypRef Expression
1 umgraf2 23993 . . 3  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 fndm 5678 . . . 4  |-  ( E  Fn  A  ->  dom  E  =  A )
32feq2d 5716 . . 3  |-  ( E  Fn  A  ->  ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
41, 3syl5ibcom 220 . 2  |-  ( V UMGrph  E  ->  ( E  Fn  A  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } ) )
54imp 429 1  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   {crab 2818    \ cdif 3473   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447   dom cdm 4999    Fn wfn 5581   -->wf 5582   ` cfv 5586    <_ cle 9625   2c2 10581   #chash 12369   UMGrph cumg 23988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5588  df-fn 5589  df-f 5590  df-umgra 23989
This theorem is referenced by:  umgrass  23995  umgran0  23996  umgrale  23997  umgraun  24004
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