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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 (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 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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