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Theorem isuspgrop 29193
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 5475 . . 3 𝑉, 𝐸⟩ ∈ V
2 eqid 2735 . . . 4 (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
3 eqid 2735 . . . 4 (iEdg‘⟨𝑉, 𝐸⟩) = (iEdg‘⟨𝑉, 𝐸⟩)
42, 3isuspgr 29184 . . 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 29039 . . 3 ((𝑉𝑊𝐸𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
76dmeqd 5919 . . 3 ((𝑉𝑊𝐸𝑋) → dom (iEdg‘⟨𝑉, 𝐸⟩) = dom 𝐸)
8 opvtxfv 29036 . . . . . 6 ((𝑉𝑊𝐸𝑋) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
98pweqd 4622 . . . . 5 ((𝑉𝑊𝐸𝑋) → 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) = 𝒫 𝑉)
109difeq1d 4135 . . . 4 ((𝑉𝑊𝐸𝑋) → (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1110rabeqdv 3449 . . 3 ((𝑉𝑊𝐸𝑋) → {𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
126, 7, 11f1eq123d 6841 . 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 279 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 206  wa 395  wcel 2106  {crab 3433  Vcvv 3478  cdif 3960  c0 4339  𝒫 cpw 4605  {csn 4631  cop 4637   class class class wbr 5148  dom cdm 5689  1-1wf1 6560  cfv 6563  cle 11294  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  USPGraphcuspgr 29180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571  df-1st 8013  df-2nd 8014  df-vtx 29030  df-iedg 29031  df-uspgr 29182
This theorem is referenced by:  uspgrsprfo  47992
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