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

Theorem umisuhgra 23406
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 23393 . . . 4  |-  Rel UMGrph
21brrelexi 4980 . . 3  |-  ( V UMGrph  E  ->  V  e.  _V )
31brrelex2i 4981 . . 3  |-  ( V UMGrph  E  ->  E  e.  _V )
42, 3jca 532 . 2  |-  ( V UMGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
5 isumgra 23394 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }
) )
6 ssrab2 3538 . . . . 5  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  <_  2 }  C_  ( ~P V  \  { (/) } )
7 fss 5668 . . . . 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 671 . . . 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 23382 . . 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 369    e. wcel 1758   {crab 2799   _Vcvv 3071    \ cdif 3426    C_ wss 3429   (/)c0 3738   ~Pcpw 3961   {csn 3978   class class class wbr 4393   dom cdm 4941   -->wf 5515   ` cfv 5519    <_ cle 9523   2c2 10475   #chash 12213   UHGrph cuhg 23378   UMGrph cumg 23391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-fun 5521  df-fn 5522  df-f 5523  df-uhgra 23379  df-umgra 23392
This theorem is referenced by:  usisuhgra  30415
  Copyright terms: Public domain W3C validator