Theorem upgrop 29163

Description: A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.)

Assertion

Ref	Expression
upgrop	⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)

Proof of Theorem upgrop

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2736	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrf 29155	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
4		fvex 6853	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
5		fvex 6853	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
6	4, 5	pm3.2i 470	. . 3 ⊢ ((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V)
7		opex 5416	. . . . 5 ⊢ ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V
8		eqid 2736	. . . . . 6 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
9		eqid 2736	. . . . . 6 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
10	8, 9	isupgr 29153	. . . . 5 ⊢ (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V → (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ↔ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩):dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)⟶{𝑝 ∈ (𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
11	7, 10	mp1i 13	. . . 4 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ↔ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩):dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)⟶{𝑝 ∈ (𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
12		opiedgfv 29076	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
13	12	dmeqd 5860	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
14		opvtxfv 29073	. . . . . . . 8 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
15	14	pweqd 4558	. . . . . . 7 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = 𝒫 (Vtx‘𝐺))
16	15	difeq1d 4065	. . . . . 6 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∖ {∅}) = (𝒫 (Vtx‘𝐺) ∖ {∅}))
17	16	rabeqdv 3404	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → {𝑝 ∈ (𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} = {𝑝 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
18	12, 13, 17	feq123d 6657	. . . 4 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩):dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)⟶{𝑝 ∈ (𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2} ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
19	11, 18	bitrd 279	. . 3 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
20	6, 19	mp1i 13	. 2 ⊢ (𝐺 ∈ UPGraph → (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}))
21	3, 20	mpbird 257	1 ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)

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  ⟶wf 6494  ‘cfv 6498   ≤ cle 11180  2c2 12236  ♯chash 14292  Vtxcvtx 29065  iEdgciedg 29066  UPGraphcupgr 29149

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-fv 6506  df-1st 7942  df-2nd 7943  df-vtx 29067  df-iedg 29068  df-upgr 29151

This theorem is referenced by:  finsumvtxdg2size  29619