MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  upgruhgr Structured version   Visualization version   GIF version

Theorem upgruhgr 27472
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 2738 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2738 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2upgrf 27456 . . 3 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4 ssrab2 4013 . . 3 {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5 fss 6617 . . 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 27430 . 2 (𝐺 ∈ UPGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
86, 7mpbird 256 1 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  {crab 3068  cdif 3884  wss 3887  c0 4256  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  dom cdm 5589  wf 6429  cfv 6433  cle 11010  2c2 12028  chash 14044  Vtxcvtx 27366  iEdgciedg 27367  UHGraphcuhgr 27426  UPGraphcupgr 27450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-uhgr 27428  df-upgr 27452
This theorem is referenced by:  umgruhgr  27474  upgrle2  27475  edglnl  27513  numedglnl  27514  usgruhgr  27553  subupgr  27654  upgrspan  27660  upgrreslem  27671  upgrres  27673  finsumvtxdg2ssteplem1  27912  finsumvtxdg2size  27917  upgrewlkle2  27973  upgredginwlk  28003  wlkiswwlks1  28232  wlkiswwlksupgr2  28242  eulerpathpr  28604  eulercrct  28606  upgracycumgr  33115  isomuspgrlem2c  45282
  Copyright terms: Public domain W3C validator