Theorem isuspgrop 29230

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	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))

Distinct variable groups:   𝐸,𝑝   𝑉,𝑝   𝑊,𝑝   𝑋,𝑝

Proof of Theorem isuspgrop

Step	Hyp	Ref	Expression
1		opex 5416	. . 3 ⊢ ⟨𝑉, 𝐸⟩ ∈ V
2		eqid 2736	. . . 4 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
3		eqid 2736	. . . 4 ⊢ (iEdg‘⟨𝑉, 𝐸⟩) = (iEdg‘⟨𝑉, 𝐸⟩)
4	2, 3	isuspgr 29221	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ V → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
5	1, 4	mp1i 13	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ (iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
6		opiedgfv 29076	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
7	6	dmeqd 5860	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom (iEdg‘⟨𝑉, 𝐸⟩) = dom 𝐸)
8		opvtxfv 29073	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
9	8	pweqd 4558	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝒫 (Vtx‘⟨𝑉, 𝐸⟩) = 𝒫 𝑉)
10	9	difeq1d 4065	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
11	10	rabeqdv 3404	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
12	6, 7, 11	f1eq123d 6772	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ((iEdg‘⟨𝑉, 𝐸⟩):dom (iEdg‘⟨𝑉, 𝐸⟩)–1-1→{𝑝 ∈ (𝒫 (Vtx‘⟨𝑉, 𝐸⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
13	5, 12	bitrd 279	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  {crab 3389  Vcvv 3429   ∖ cdif 3886  ∅c0 4273  𝒫 cpw 4541  {csn 4567  ⟨cop 4573   class class class wbr 5085  dom cdm 5631  –1-1→wf1 6495  ‘cfv 6498   ≤ cle 11180  2c2 12236  ♯chash 14292  Vtxcvtx 29065  iEdgciedg 29066  USPGraphcuspgr 29217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fv 6506  df-1st 7942  df-2nd 7943  df-vtx 29067  df-iedg 29068  df-uspgr 29219

This theorem is referenced by:  uspgrsprfo  48624