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Theorem upgruhgr 29133
Description: An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.)
Assertion
Ref Expression
upgruhgr (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)

Proof of Theorem upgruhgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2734 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2upgrf 29117 . . 3 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4 ssrab2 4089 . . 3 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5 fss 6752 . . 3 (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
63, 4, 5sylancl 586 . 2 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
71, 2isuhgr 29091 . 2 (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
86, 7mpbird 257 1 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  {crab 3432  cdif 3959  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  dom cdm 5688  wf 6558  cfv 6562  cle 11293  2c2 12318  chash 14365  Vtxcvtx 29027  iEdgciedg 29028  UHGraphcuhgr 29087  UPGraphcupgr 29111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-uhgr 29089  df-upgr 29113
This theorem is referenced by:  umgruhgr  29135  upgrle2  29136  edglnl  29174  numedglnl  29175  uspgruhgr  29215  usgruhgr  29217  subupgr  29318  upgrspan  29324  upgrreslem  29335  upgrres  29337  finsumvtxdg2ssteplem1  29577  finsumvtxdg2size  29582  upgrewlkle2  29638  upgredginwlk  29668  wlkiswwlks1  29896  wlkiswwlksupgr2  29906  eulerpathpr  30268  eulercrct  30270  upgracycumgr  35137  isubgrupgr  47793
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