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Theorem upgruhgr 29193
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 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2737 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2upgrf 29177 . . 3 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4 ssrab2 4034 . . 3 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5 fss 6688 . . 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 587 . 2 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
71, 2isuhgr 29151 . 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 2114  {crab 3401  cdif 3900  wss 3903  c0 4287  𝒫 cpw 4556  {csn 4582   class class class wbr 5100  dom cdm 5634  wf 6498  cfv 6502  cle 11181  2c2 12214  chash 14267  Vtxcvtx 29087  iEdgciedg 29088  UHGraphcuhgr 29147  UPGraphcupgr 29171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5255
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fv 6510  df-uhgr 29149  df-upgr 29173
This theorem is referenced by:  umgruhgr  29195  upgrle2  29196  edglnl  29234  numedglnl  29235  uspgruhgr  29275  usgruhgr  29277  subupgr  29378  upgrspan  29384  upgrreslem  29395  upgrres  29397  finsumvtxdg2ssteplem1  29637  finsumvtxdg2size  29642  upgrewlkle2  29698  upgredginwlk  29727  wlkiswwlks1  29958  wlkiswwlksupgr2  29968  eulerpathpr  30333  eulercrct  30335  upgracycumgr  35375  isubgrupgr  48259
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