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Theorem umisuhgra 21315
Description: An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
Assertion
Ref Expression
umisuhgra  |-  ( V UMGrph  E  ->  V UHGrph  E )

Proof of Theorem umisuhgra
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 relumgra 21302 . . . 4  |-  Rel UMGrph
21brrelexi 4877 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
31brrelex2i 4878 . . 3  |-  ( V UMGrph  E  ->  E  e.  _V )
42, 3jca 519 . 2  |-  ( V UMGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
5 isumgra 21303 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
6 ssrab2 3388 . . . . 5  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_  ( ~P V  \  { (/) } )
7 fss 5558 . . . . 5  |-  ( ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }  /\  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_  ( ~P V  \  { (/) } ) )  ->  E : dom  E --> ( ~P V  \  { (/) } ) )
86, 7mpan2 653 . . . 4  |-  ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  E : dom  E --> ( ~P V  \  { (/)
} ) )
95, 8syl6bi 220 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
10 isuhgra 21291 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
119, 10sylibrd 226 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  ->  V UHGrph  E ) )
124, 11mpcom 34 1  |-  ( V UMGrph  E  ->  V UHGrph  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   class class class wbr 4172   dom cdm 4837   -->wf 5409   ` cfv 5413    <_ cle 9077   2c2 10005   #chash 11573   UHGrph cuhg 21287   UMGrph cumg 21300
This theorem is referenced by:  usisuhgra  28033
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 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-uhgra 21288  df-umgra 21301
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