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Theorem upgrun 26817
 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 (𝐸 ∪ 𝐹) 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 26785 . . . . 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 2826 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘𝐻)
8 upgrun.f . . . . . . 7 𝐹 = (iEdg‘𝐻)
97, 8upgrf 26785 . . . . . 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 2832 . . . . . . . . 9 (𝜑𝑉 = (Vtx‘𝐻))
1312pweqd 4547 . . . . . . . 8 (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
1413difeq1d 4102 . . . . . . 7 (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
1514rabeqdv 3490 . . . . . 6 (𝜑 → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1615feq3d 6498 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1710, 16mpbird 258 . . . 4 (𝜑𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18 upgrun.i . . . 4 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
195, 17, 18fun2d 6539 . . 3 (𝜑 → (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
20 upgrun.un . . . 4 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
2120dmeqd 5773 . . . . 5 (𝜑 → dom (iEdg‘𝑈) = dom (𝐸𝐹))
22 dmun 5778 . . . . 5 dom (𝐸𝐹) = (dom 𝐸 ∪ dom 𝐹)
2321, 22syl6eq 2877 . . . 4 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
24 upgrun.v . . . . . . 7 (𝜑 → (Vtx‘𝑈) = 𝑉)
2524pweqd 4547 . . . . . 6 (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
2625difeq1d 4102 . . . . 5 (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2726rabeqdv 3490 . . . 4 (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
2820, 23, 27feq123d 6500 . . 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 258 . 2 (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
30 upgrun.u . . 3 (𝜑𝑈𝑊)
31 eqid 2826 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
32 eqid 2826 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
3331, 32isupgr 26783 . . 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 258 1 (𝜑𝑈 ∈ UPGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   = wceq 1530   ∈ wcel 2107  {crab 3147   ∖ cdif 3937   ∪ cun 3938   ∩ cin 3939  ∅c0 4295  𝒫 cpw 4542  {csn 4564   class class class wbr 5063  dom cdm 5554  ⟶wf 6348  ‘cfv 6352   ≤ cle 10665  2c2 11681  ♯chash 13680  Vtxcvtx 26695  iEdgciedg 26696  UPGraphcupgr 26779 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-upgr 26781 This theorem is referenced by:  upgrunop  26818  uspgrun  26884
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