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Theorem umgraf 21314
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 21313 . . 3  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 fndm 5511 . . . 4  |-  ( E  Fn  A  ->  dom  E  =  A )
32feq2d 5548 . . 3  |-  ( E  Fn  A  ->  ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
41, 3syl5ibcom 212 . 2  |-  ( V UMGrph  E  ->  ( E  Fn  A  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } ) )
54imp 419 1  |-  ( ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   {crab 2678    \ cdif 3285   (/)c0 3596   ~Pcpw 3767   {csn 3782   class class class wbr 4180   dom cdm 4845    Fn wfn 5416   -->wf 5417   ` cfv 5421    <_ cle 9085   2c2 10013   #chash 11581   UMGrph cumg 21308
This theorem is referenced by:  umgrass  21315  umgran0  21316  umgrale  21317  umgraun  21324
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-fun 5423  df-fn 5424  df-f 5425  df-umgra 21309
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