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Theorem umgraf 24616
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 24615 . . 3  |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  <_  2 } )
2 fndm 5617 . . . 4  |-  ( E  Fn  A  ->  dom  E  =  A )
32feq2d 5657 . . 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 427 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 367   {crab 2757    \ cdif 3410   (/)c0 3737   ~Pcpw 3954   {csn 3971   class class class wbr 4394   dom cdm 4942    Fn wfn 5520   -->wf 5521   ` cfv 5525    <_ cle 9579   2c2 10546   #chash 12359   UMGrph cumg 24610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-fun 5527  df-fn 5528  df-f 5529  df-umgra 24611
This theorem is referenced by:  umgrass  24617  umgran0  24618  umgrale  24619  umgraun  24626
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