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Theorem isumgra 21340
 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 448 . . . 4
21dmeqd 5064 . . . 4
31, 2feq12d 5574 . . 3
4 simpl 444 . . . . . 6
54pweqd 3796 . . . . 5
65difeq1d 3456 . . . 4
7 rabeq 2942 . . . 4
8 feq3 5570 . . . 4
96, 7, 83syl 19 . . 3
103, 9bitrd 245 . 2
11 df-umgra 21338 . 2 UMGrph
1210, 11brabga 4461 1 UMGrph
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  crab 2701   cdif 3309  c0 3620  cpw 3791  csn 3806   class class class wbr 4204   cdm 4870  wf 5442  cfv 5446   cle 9111  c2 10039  chash 11608   UMGrph cumg 21337 This theorem is referenced by:  wrdumgra  21341  umgraf2  21342  umgrares  21349  umgra0  21350  umgra1  21351  umisuhgra  21352  umgraun  21353  uslisumgra  21376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-umgra 21338
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