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Theorem umisuhgra 25856
 Description: An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
Assertion
Ref Expression
umisuhgra (𝑉 UMGrph 𝐸𝑉 UHGrph 𝐸)

Proof of Theorem umisuhgra
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 relumgra 25843 . . . 4 Rel UMGrph
21brrelexi 5082 . . 3 (𝑉 UMGrph 𝐸𝑉 ∈ V)
31brrelex2i 5083 . . 3 (𝑉 UMGrph 𝐸𝐸 ∈ V)
42, 3jca 553 . 2 (𝑉 UMGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
5 isumgra 25844 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UMGrph 𝐸𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
6 ssrab2 3650 . . . . 5 {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅})
7 fss 5969 . . . . 5 ((𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅})) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
86, 7mpan2 703 . . . 4 (𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
95, 8syl6bi 242 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UMGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
10 isuhgra 25827 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
119, 10sylibrd 248 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UMGrph 𝐸𝑉 UHGrph 𝐸))
124, 11mpcom 37 1 (𝑉 UMGrph 𝐸𝑉 UHGrph 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979   UHGrph cuhg 25819   UMGrph cumg 25841 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-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 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-br 4584  df-opab 4644  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-uhgra 25821  df-umgra 25842 This theorem is referenced by:  usisuhgra  25896
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