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Theorem uhgredgn0 26363
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 26284 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 eqid 2799 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2799 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3uhgrf 26297 . . . 4 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
54frnd 6263 . . 3 (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
61, 5syl5eqss 3845 . 2 (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
76sselda 3798 1 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  cdif 3766  c0 4115  𝒫 cpw 4349  {csn 4368  dom cdm 5312  ran crn 5313  cfv 6101  Vtxcvtx 26231  iEdgciedg 26232  Edgcedg 26282  UHGraphcuhgr 26291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-edg 26283  df-uhgr 26293
This theorem is referenced by:  edguhgr  26364  uhgredgss  26366  uhgrvd00  26784
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