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Theorem uslgraun 25903
 Description: The union of two simple graphs with loops (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are (simple) graphs (with loops), then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a multigraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices), analogous to umgraun 25857. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Hypotheses
Ref Expression
uslgraun.e (𝜑𝐸 Fn 𝐴)
uslgraun.f (𝜑𝐹 Fn 𝐵)
uslgraun.i (𝜑 → (𝐴𝐵) = ∅)
uslgraun.ge (𝜑𝑉 USLGrph 𝐸)
uslgraun.gf (𝜑𝑉 USLGrph 𝐹)
Assertion
Ref Expression
uslgraun (𝜑𝑉 UMGrph (𝐸𝐹))

Proof of Theorem uslgraun
StepHypRef Expression
1 uslgraun.e . 2 (𝜑𝐸 Fn 𝐴)
2 uslgraun.f . 2 (𝜑𝐹 Fn 𝐵)
3 uslgraun.i . 2 (𝜑 → (𝐴𝐵) = ∅)
4 uslgraun.ge . . 3 (𝜑𝑉 USLGrph 𝐸)
5 uslisumgra 25893 . . 3 (𝑉 USLGrph 𝐸𝑉 UMGrph 𝐸)
64, 5syl 17 . 2 (𝜑𝑉 UMGrph 𝐸)
7 uslgraun.gf . . 3 (𝜑𝑉 USLGrph 𝐹)
8 uslisumgra 25893 . . 3 (𝑉 USLGrph 𝐹𝑉 UMGrph 𝐹)
97, 8syl 17 . 2 (𝜑𝑉 UMGrph 𝐹)
101, 2, 3, 6, 9umgraun 25857 1 (𝜑𝑉 UMGrph (𝐸𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538   ∩ cin 3539  ∅c0 3874   class class class wbr 4583   Fn wfn 5799   UMGrph cumg 25841   USLGrph cuslg 25858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-umgra 25842  df-uslgra 25861 This theorem is referenced by: (None)
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