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Theorem isuhgrop 40297
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.) (Revised by AV, 9-Oct-2020.)
Assertion
Ref Expression
isuhgrop ((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Proof of Theorem isuhgrop
StepHypRef Expression
1 opex 4853 . . 3 𝑉, 𝐸⟩ ∈ V
2 eqid 2609 . . . 4 (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
3 eqid 2609 . . . 4 (iEdg‘⟨𝑉, 𝐸⟩) = (iEdg‘⟨𝑉, 𝐸⟩)
42, 3isuhgr 40284 . . 3 (⟨𝑉, 𝐸⟩ ∈ V → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)⟶(𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅})))
51, 4mp1i 13 . 2 ((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)⟶(𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅})))
6 opiedgfv 40242 . . 3 ((𝑉𝑊𝐸𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
76dmeqd 5235 . . 3 ((𝑉𝑊𝐸𝑋) → dom (iEdg‘⟨𝑉, 𝐸⟩) = dom 𝐸)
8 opvtxfv 40239 . . . . 5 ((𝑉𝑊𝐸𝑋) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
98pweqd 4112 . . . 4 ((𝑉𝑊𝐸𝑋) → 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) = 𝒫 𝑉)
109difeq1d 3688 . . 3 ((𝑉𝑊𝐸𝑋) → (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
116, 7, 10feq123d 5933 . 2 ((𝑉𝑊𝐸𝑋) → ((iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)⟶(𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
125, 11bitrd 266 1 ((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  Vcvv 3172  cdif 3536  c0 3873  𝒫 cpw 4107  {csn 4124  cop 4130  dom cdm 5028  wf 5786  cfv 5790  Vtxcvtx 40231  iEdgciedg 40232   UHGraph cuhgr 40280
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-8 1978  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-pow 4764  ax-pr 4828  ax-un 6824
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-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  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-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-1st 7036  df-2nd 7037  df-vtx 40233  df-iedg 40234  df-uhgr 40282
This theorem is referenced by: (None)
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