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Theorem isuspgrop 27276
Description: The property of being an undirected simple pseudograph 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, 25-Nov-2021.)
Assertion
Ref Expression
isuspgrop ((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
Distinct variable groups:   𝐸,𝑝   𝑉,𝑝   𝑊,𝑝   𝑋,𝑝

Proof of Theorem isuspgrop
StepHypRef Expression
1 opex 5363 . . 3 𝑉, 𝐸⟩ ∈ V
2 eqid 2738 . . . 4 (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
3 eqid 2738 . . . 4 (iEdg‘⟨𝑉, 𝐸⟩) = (iEdg‘⟨𝑉, 𝐸⟩)
42, 3isuspgr 27267 . . 3 (⟨𝑉, 𝐸⟩ ∈ V → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
51, 4mp1i 13 . 2 ((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
6 opiedgfv 27122 . . 3 ((𝑉𝑊𝐸𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
76dmeqd 5789 . . 3 ((𝑉𝑊𝐸𝑋) → dom (iEdg‘⟨𝑉, 𝐸⟩) = dom 𝐸)
8 opvtxfv 27119 . . . . . 6 ((𝑉𝑊𝐸𝑋) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
98pweqd 4547 . . . . 5 ((𝑉𝑊𝐸𝑋) → 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) = 𝒫 𝑉)
109difeq1d 4051 . . . 4 ((𝑉𝑊𝐸𝑋) → (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1110rabeqdv 3408 . . 3 ((𝑉𝑊𝐸𝑋) → {𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
126, 7, 11f1eq123d 6672 . 2 ((𝑉𝑊𝐸𝑋) → ((iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ 𝐸:dom 𝐸1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
135, 12bitrd 282 1 ((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2111  {crab 3066  Vcvv 3421  cdif 3878  c0 4252  𝒫 cpw 4528  {csn 4556  cop 4562   class class class wbr 5068  dom cdm 5566  1-1wf1 6395  cfv 6398  cle 10893  2c2 11910  chash 13921  Vtxcvtx 27111  iEdgciedg 27112  USPGraphcuspgr 27263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-sbc 3710  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fv 6406  df-1st 7780  df-2nd 7781  df-vtx 27113  df-iedg 27114  df-uspgr 27265
This theorem is referenced by:  uspgrsprfo  45012
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