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Theorem umgraun 24455
 Description: The union of two (undirected) multigraphs (with the same vertex set): If and are graphs, then is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
Hypotheses
Ref Expression
umgraun.e
umgraun.f
umgraun.i
umgraun.ge UMGrph
umgraun.gf UMGrph
Assertion
Ref Expression
umgraun UMGrph

Proof of Theorem umgraun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 umgraun.ge . . . . 5 UMGrph
2 umgraun.e . . . . 5
3 umgraf 24445 . . . . 5 UMGrph
41, 2, 3syl2anc 661 . . . 4
5 umgraun.gf . . . . 5 UMGrph
6 umgraun.f . . . . 5
7 umgraf 24445 . . . . 5 UMGrph
85, 6, 7syl2anc 661 . . . 4
9 umgraun.i . . . 4
10 fun2 5755 . . . 4
114, 8, 9, 10syl21anc 1227 . . 3
12 fdm 5741 . . . . 5
1311, 12syl 16 . . . 4
1413feq2d 5724 . . 3
1511, 14mpbird 232 . 2
16 relumgra 24441 . . . 4 UMGrph
17 releldm 5245 . . . 4 UMGrph UMGrph UMGrph
1816, 1, 17sylancr 663 . . 3 UMGrph
1916brrelex2i 5050 . . . . 5 UMGrph
201, 19syl 16 . . . 4
2116brrelex2i 5050 . . . . 5 UMGrph
225, 21syl 16 . . . 4
23 unexg 6600 . . . 4
2420, 22, 23syl2anc 661 . . 3
25 isumgra 24442 . . 3 UMGrph UMGrph
2618, 24, 25syl2anc 661 . 2 UMGrph
2715, 26mpbird 232 1 UMGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wceq 1395   wcel 1819  crab 2811  cvv 3109   cdif 3468   cun 3469   cin 3470  c0 3793  cpw 4015  csn 4032   class class class wbr 4456   cdm 5008   wrel 5013   wfn 5589  wf 5590  cfv 5594   cle 9646  c2 10606  chash 12408   UMGrph cumg 24439 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598  df-umgra 24440 This theorem is referenced by:  uslgraun  24501  eupap1  25103
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