Theorem upgrop 27367

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 2738	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2738	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrf 27359	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
4		fvex 6769	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
5		fvex 6769	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
6	4, 5	pm3.2i 470	. . 3 ⊢ ((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V)
7		opex 5373	. . . . 5 ⊢ ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V
8		eqid 2738	. . . . . 6 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
9		eqid 2738	. . . . . 6 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
10	8, 9	isupgr 27357	. . . . 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 27280	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
13	12	dmeqd 5803	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
14		opvtxfv 27277	. . . . . . . 8 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
15	14	pweqd 4549	. . . . . . 7 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = 𝒫 (Vtx‘𝐺))
16	15	difeq1d 4052	. . . . . 6 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∖ {∅}) = (𝒫 (Vtx‘𝐺) ∖ {∅}))
17	16	rabeqdv 3409	. . . . 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 6573	. . . 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 278	. . 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 256	1 ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∖ cdif 3880  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ⟨cop 4564   class class class wbr 5070  dom cdm 5580  ⟶wf 6414  ‘cfv 6418   ≤ cle 10941  2c2 11958  ♯chash 13972  Vtxcvtx 27269  iEdgciedg 27270  UPGraphcupgr 27353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-1st 7804  df-2nd 7805  df-vtx 27271  df-iedg 27272  df-upgr 27355

This theorem is referenced by:  finsumvtxdg2size  27820