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Theorem ushgruhgr 29086
Description: An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
Assertion
Ref Expression
ushgruhgr (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)

Proof of Theorem ushgruhgr
StepHypRef Expression
1 eqid 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2737 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2ushgrf 29080 . . 3 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 f1f 6804 . . 3 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
53, 4syl 17 . 2 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
61, 2isuhgr 29077 . 2 (𝐺 ∈ USHGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
75, 6mpbird 257 1 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cdif 3948  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  wf 6557  1-1wf1 6558  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073  USHGraphcushgr 29074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-uhgr 29075  df-ushgr 29076
This theorem is referenced by:  ushgrun  29093  ushgrunop  29094  ushgredgedg  29246  ushgredgedgloop  29248  ushggricedg  47896
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