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Theorem ushgrauhgra 25832
 Description: An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.)
Assertion
Ref Expression
ushgrauhgra (𝑉 USHGrph 𝐸𝑉 UHGrph 𝐸)

Proof of Theorem ushgrauhgra
StepHypRef Expression
1 relushgra 25824 . . . 4 Rel USHGrph
21brrelexi 5082 . . 3 (𝑉 USHGrph 𝐸𝑉 ∈ V)
31brrelex2i 5083 . . 3 (𝑉 USHGrph 𝐸𝐸 ∈ V)
42, 3jca 553 . 2 (𝑉 USHGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
5 isushgra 25830 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USHGrph 𝐸𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
6 f1f 6014 . . . 4 (𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
7 isuhgra 25827 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
86, 7syl5ibr 235 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}) → 𝑉 UHGrph 𝐸))
95, 8sylbid 229 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USHGrph 𝐸𝑉 UHGrph 𝐸))
104, 9mpcom 37 1 (𝑉 USHGrph 𝐸𝑉 UHGrph 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  –1-1→wf1 5801   UHGrph cuhg 25819   USHGrph cushg 25820 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-f1 5809  df-uhgra 25821  df-ushgra 25822 This theorem is referenced by: (None)
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