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Theorem uhgraop 25599
Description: The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.)
Assertion
Ref Expression
uhgraop ((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGrph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Proof of Theorem uhgraop
StepHypRef Expression
1 df-br 4578 . 2 (𝑉 UHGrph 𝐸 ↔ ⟨𝑉, 𝐸⟩ ∈ UHGrph )
2 isuhgra 25593 . 2 ((𝑉𝑊𝐸𝑋) → (𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
31, 2syl5bbr 272 1 ((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGrph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  cdif 3536  c0 3873  𝒫 cpw 4107  {csn 4124  cop 4130   class class class wbr 4577  dom cdm 5028  wf 5786   UHGrph cuhg 25585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794  df-uhgra 25587
This theorem is referenced by: (None)
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