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Theorem uhgredgn0 26214
Description: An edge of a hypergraph is a nonempty subset of vertices. (Contributed by AV, 28-Nov-2020.)
Assertion
Ref Expression
uhgredgn0 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Proof of Theorem uhgredgn0
StepHypRef Expression
1 edgval 26132 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 eqid 2752 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2752 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3uhgrf 26148 . . . 4 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
5 frn 6206 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
64, 5syl 17 . . 3 (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
71, 6syl5eqss 3782 . 2 (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
87sselda 3736 1 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2131  cdif 3704  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313  dom cdm 5258  ran crn 5259  wf 6037  cfv 6041  Vtxcvtx 26065  iEdgciedg 26066  Edgcedg 26130  UHGraphcuhgr 26142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-edg 26131  df-uhgr 26144
This theorem is referenced by:  edguhgr  26215  uhgredgss  26217  uhgrvd00  26632
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