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Theorem upgrun 29135
Description: The union 𝑈 of two pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)
Hypotheses
Ref Expression
upgrun.g (𝜑𝐺 ∈ UPGraph)
upgrun.h (𝜑𝐻 ∈ UPGraph)
upgrun.e 𝐸 = (iEdg‘𝐺)
upgrun.f 𝐹 = (iEdg‘𝐻)
upgrun.vg 𝑉 = (Vtx‘𝐺)
upgrun.vh (𝜑 → (Vtx‘𝐻) = 𝑉)
upgrun.i (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
upgrun.u (𝜑𝑈𝑊)
upgrun.v (𝜑 → (Vtx‘𝑈) = 𝑉)
upgrun.un (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
Assertion
Ref Expression
upgrun (𝜑𝑈 ∈ UPGraph)

Proof of Theorem upgrun
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 upgrun.g . . . . 5 (𝜑𝐺 ∈ UPGraph)
2 upgrun.vg . . . . . 6 𝑉 = (Vtx‘𝐺)
3 upgrun.e . . . . . 6 𝐸 = (iEdg‘𝐺)
42, 3upgrf 29103 . . . . 5 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
51, 4syl 17 . . . 4 (𝜑𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
6 upgrun.h . . . . . 6 (𝜑𝐻 ∈ UPGraph)
7 eqid 2737 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘𝐻)
8 upgrun.f . . . . . . 7 𝐹 = (iEdg‘𝐻)
97, 8upgrf 29103 . . . . . 6 (𝐻 ∈ UPGraph → 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
106, 9syl 17 . . . . 5 (𝜑𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
11 upgrun.vh . . . . . . . . . 10 (𝜑 → (Vtx‘𝐻) = 𝑉)
1211eqcomd 2743 . . . . . . . . 9 (𝜑𝑉 = (Vtx‘𝐻))
1312pweqd 4617 . . . . . . . 8 (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
1413difeq1d 4125 . . . . . . 7 (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
1514rabeqdv 3452 . . . . . 6 (𝜑 → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1615feq3d 6723 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1710, 16mpbird 257 . . . 4 (𝜑𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18 upgrun.i . . . 4 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
195, 17, 18fun2d 6772 . . 3 (𝜑 → (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
20 upgrun.un . . . 4 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
2120dmeqd 5916 . . . . 5 (𝜑 → dom (iEdg‘𝑈) = dom (𝐸𝐹))
22 dmun 5921 . . . . 5 dom (𝐸𝐹) = (dom 𝐸 ∪ dom 𝐹)
2321, 22eqtrdi 2793 . . . 4 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
24 upgrun.v . . . . . . 7 (𝜑 → (Vtx‘𝑈) = 𝑉)
2524pweqd 4617 . . . . . 6 (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
2625difeq1d 4125 . . . . 5 (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2726rabeqdv 3452 . . . 4 (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
2820, 23, 27feq123d 6725 . . 3 (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
2919, 28mpbird 257 . 2 (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
30 upgrun.u . . 3 (𝜑𝑈𝑊)
31 eqid 2737 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
32 eqid 2737 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
3331, 32isupgr 29101 . . 3 (𝑈𝑊 → (𝑈 ∈ UPGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
3430, 33syl 17 . 2 (𝜑 → (𝑈 ∈ UPGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
3529, 34mpbird 257 1 (𝜑𝑈 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {crab 3436  cdif 3948  cun 3949  cin 3950  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  dom cdm 5685  wf 6557  cfv 6561  cle 11296  2c2 12321  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  UPGraphcupgr 29097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-upgr 29099
This theorem is referenced by:  upgrunop  29136  uspgrun  29205
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