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Theorem upgruhgr 26801
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 2826 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2826 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2upgrf 26785 . . 3 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4 ssrab2 4060 . . 3 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5 fss 6524 . . 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 26759 . 2 (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
86, 7mpbird 258 1 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  {crab 3147  cdif 3937  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564   class class class wbr 5063  dom cdm 5554  wf 6348  cfv 6352  cle 10665  2c2 11681  chash 13680  Vtxcvtx 26695  iEdgciedg 26696  UHGraphcuhgr 26755  UPGraphcupgr 26779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-nul 5207
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-uhgr 26757  df-upgr 26781
This theorem is referenced by:  umgruhgr  26803  upgrle2  26804  edglnl  26842  numedglnl  26843  usgruhgr  26882  subupgr  26983  upgrspan  26989  upgrreslem  27000  upgrres  27002  finsumvtxdg2ssteplem1  27241  finsumvtxdg2size  27246  upgrewlkle2  27302  upgredginwlk  27331  wlkiswwlks1  27559  wlkiswwlksupgr2  27569  eulerpathpr  27933  eulercrct  27935  upgracycumgr  32284  isomuspgrlem2c  43827
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