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Theorem upgrun 25784
 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 25753 . . . . 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 2610 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘𝐻)
8 upgrun.f . . . . . . 7 𝐹 = (iEdg‘𝐻)
97, 8upgrf 25753 . . . . . 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 2616 . . . . . . . . 9 (𝜑𝑉 = (Vtx‘𝐻))
1312pweqd 4113 . . . . . . . 8 (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
1413difeq1d 3689 . . . . . . 7 (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
1514rabeqdv 3167 . . . . . 6 (𝜑 → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
1615feq3d 5945 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
1710, 16mpbird 246 . . . 4 (𝜑𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
18 upgrun.i . . . 4 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
195, 17, 18fun2d 5981 . . 3 (𝜑 → (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
20 upgrun.un . . . 4 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
2120dmeqd 5248 . . . . 5 (𝜑 → dom (iEdg‘𝑈) = dom (𝐸𝐹))
22 dmun 5253 . . . . 5 dom (𝐸𝐹) = (dom 𝐸 ∪ dom 𝐹)
2321, 22syl6eq 2660 . . . 4 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
24 upgrun.v . . . . . . 7 (𝜑 → (Vtx‘𝑈) = 𝑉)
2524pweqd 4113 . . . . . 6 (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
2625difeq1d 3689 . . . . 5 (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2726rabeqdv 3167 . . . 4 (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
2820, 23, 27feq123d 5947 . . 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 246 . 2 (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
30 upgrun.u . . 3 (𝜑𝑈𝑊)
31 eqid 2610 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
32 eqid 2610 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
3331, 32isupgr 25751 . . 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 246 1 (𝜑𝑈 ∈ UPGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-upgr 25749 This theorem is referenced by:  upgrunop  25785  uspgrun  40415
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