MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  umisuhgra Structured version   Unicode version

Theorem umisuhgra 24529
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 24516 . . . 4  |-  Rel UMGrph
21brrelexi 5029 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
31brrelex2i 5030 . . 3  |-  ( V UMGrph  E  ->  E  e.  _V )
42, 3jca 530 . 2  |-  ( V UMGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
5 isumgra 24517 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
6 ssrab2 3571 . . . . 5  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_  ( ~P V  \  { (/) } )
7 fss 5721 . . . . 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 669 . . . 4  |-  ( E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  ->  E : dom  E --> ( ~P V  \  { (/)
} ) )
95, 8syl6bi 228 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
10 isuhgra 24500 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UHGrph  E  <->  E : dom  E --> ( ~P V  \  { (/) } ) ) )
119, 10sylibrd 234 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  ->  V UHGrph  E ) )
124, 11mpcom 36 1  |-  ( V UMGrph  E  ->  V UHGrph  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   class class class wbr 4439   dom cdm 4988   -->wf 5566   ` cfv 5570    <_ cle 9618   2c2 10581   #chash 12387   UHGrph cuhg 24492   UMGrph cumg 24514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574  df-uhgra 24494  df-umgra 24515
This theorem is referenced by:  usisuhgra  24569
  Copyright terms: Public domain W3C validator