Theorem upgrop 29083

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 2733	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2733	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrf 29075	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
4		fvex 6844	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
5		fvex 6844	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
6	4, 5	pm3.2i 470	. . 3 ⊢ ((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V)
7		opex 5409	. . . . 5 ⊢ ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V
8		eqid 2733	. . . . . 6 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
9		eqid 2733	. . . . . 6 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
10	8, 9	isupgr 29073	. . . . 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 28996	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
13	12	dmeqd 5852	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
14		opvtxfv 28993	. . . . . . . 8 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
15	14	pweqd 4568	. . . . . . 7 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = 𝒫 (Vtx‘𝐺))
16	15	difeq1d 4076	. . . . . 6 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∖ {∅}) = (𝒫 (Vtx‘𝐺) ∖ {∅}))
17	16	rabeqdv 3412	. . . . 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 6648	. . . 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 2113  {crab 3397  Vcvv 3438   ∖ cdif 3896  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ⟨cop 4583   class class class wbr 5095  dom cdm 5621  ⟶wf 6485  ‘cfv 6489   ≤ cle 11157  2c2 12190  ♯chash 14247  Vtxcvtx 28985  iEdgciedg 28986  UPGraphcupgr 29069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-1st 7930  df-2nd 7931  df-vtx 28987  df-iedg 28988  df-upgr 29071

This theorem is referenced by:  finsumvtxdg2size  29540