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Theorem isuspgrop 29094
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 5462 . . 3 𝑉, 𝐸⟩ ∈ V
2 eqid 2726 . . . 4 (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
3 eqid 2726 . . . 4 (iEdg‘⟨𝑉, 𝐸⟩) = (iEdg‘⟨𝑉, 𝐸⟩)
42, 3isuspgr 29085 . . 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 28940 . . 3 ((𝑉𝑊𝐸𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
76dmeqd 5904 . . 3 ((𝑉𝑊𝐸𝑋) → dom (iEdg‘⟨𝑉, 𝐸⟩) = dom 𝐸)
8 opvtxfv 28937 . . . . . 6 ((𝑉𝑊𝐸𝑋) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
98pweqd 4614 . . . . 5 ((𝑉𝑊𝐸𝑋) → 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) = 𝒫 𝑉)
109difeq1d 4117 . . . 4 ((𝑉𝑊𝐸𝑋) → (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
1110rabeqdv 3435 . . 3 ((𝑉𝑊𝐸𝑋) → {𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
126, 7, 11f1eq123d 6827 . 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 278 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 205  wa 394  wcel 2099  {crab 3419  Vcvv 3462  cdif 3943  c0 4322  𝒫 cpw 4597  {csn 4623  cop 4629   class class class wbr 5145  dom cdm 5674  1-1wf1 6543  cfv 6546  cle 11290  2c2 12313  chash 14342  Vtxcvtx 28929  iEdgciedg 28930  USPGraphcuspgr 29081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fv 6554  df-1st 7995  df-2nd 7996  df-vtx 28931  df-iedg 28932  df-uspgr 29083
This theorem is referenced by:  uspgrsprfo  47561
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