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Theorem umgraf 21031
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 21030 . . 3  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 fndm 5448 . . . 4  |-  ( E  Fn  A  ->  dom  E  =  A )
32feq2d 5485 . . 3  |-  ( E  Fn  A  ->  ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  <->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
41, 3syl5ibcom 211 . 2  |-  ( V UMGrph  E  ->  ( E  Fn  A  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } ) )
54imp 418 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 358   {crab 2632    \ cdif 3235   (/)c0 3543   ~Pcpw 3714   {csn 3729   class class class wbr 4125   dom cdm 4792    Fn wfn 5353   -->wf 5354   ` cfv 5358    <_ cle 9015   2c2 9942   #chash 11505   UMGrph cumg 21025
This theorem is referenced by:  umgrass  21032  umgran0  21033  umgrale  21034  umgraun  21041  uslgraun  21072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-fun 5360  df-fn 5361  df-f 5362  df-umgra 21026
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