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Theorem ushgruhgr 26865
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 2801 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2801 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2ushgrf 26859 . . 3 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 f1f 6553 . . 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 26856 . 2 (𝐺 ∈ USHGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
75, 6mpbird 260 1 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  cdif 3881  c0 4246  𝒫 cpw 4500  {csn 4528  dom cdm 5523  wf 6324  1-1wf1 6325  cfv 6328  Vtxcvtx 26792  iEdgciedg 26793  UHGraphcuhgr 26852  USHGraphcushgr 26853
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336  df-uhgr 26854  df-ushgr 26855
This theorem is referenced by:  ushgrun  26872  ushgrunop  26873  ushgredgedg  27022  ushgredgedgloop  27024  ushrisomgr  44346
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