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Theorem ushgruhgr 25894
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 2621 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2621 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2ushgrf 25888 . . 3 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 f1f 6068 . . 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 25885 . 2 (𝐺 ∈ USHGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
75, 6mpbird 247 1 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cdif 3557  c0 3897  𝒫 cpw 4136  {csn 4155  dom cdm 5084  wf 5853  1-1wf1 5854  cfv 5857  Vtxcvtx 25808  iEdgciedg 25809   UHGraph cuhgr 25881   USHGraph cushgr 25882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fv 5865  df-uhgr 25883  df-ushgr 25884
This theorem is referenced by:  ushgrun  25901  ushgrunop  25902  ushgredgedg  26048  ushgredgedgloop  26050
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