Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isumgra Unicode version

Theorem isumgra 23867
 Description: The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
isumgra UMGrph
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem isumgra
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . 4
21dmeqd 4881 . . . 4
31, 2feq12d 5381 . . 3
4 simpl 443 . . . . . 6
54pweqd 3630 . . . . 5
65difeq1d 3293 . . . 4
7 rabeq 2782 . . . 4
8 feq3 5377 . . . 4
96, 7, 83syl 18 . . 3
103, 9bitrd 244 . 2
11 df-umgra 23863 . 2 UMGrph
1210, 11brabga 4279 1 UMGrph
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623   wcel 1684  crab 2547   cdif 3149  c0 3455  cpw 3625  csn 3640   class class class wbr 4023   cdm 4689  wf 5251  cfv 5255   cle 8868  c2 9795  chash 11337   UMGrph cumg 23860 This theorem is referenced by:  wrdumgra  23868  umgraf2  23869  umgrares  23876  umgra0  23877  umgra1  23878  umgraun  23879  uslisumgra  28112  uslgraun  28120 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-umgra 23863
 Copyright terms: Public domain W3C validator