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Theorem upgrun 26234
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 26202 . . . . 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 2771 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘𝐻)
8 upgrun.f . . . . . . 7 𝐹 = (iEdg‘𝐻)
97, 8upgrf 26202 . . . . . 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 2777 . . . . . . . . 9 (𝜑𝑉 = (Vtx‘𝐻))
1312pweqd 4303 . . . . . . . 8 (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
1413difeq1d 3878 . . . . . . 7 (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
1514rabeqdv 3344 . . . . . 6 (𝜑 → {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1615feq3d 6171 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1710, 16mpbird 247 . . . 4 (𝜑𝐹:dom 𝐹⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18 upgrun.i . . . 4 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
195, 17, 18fun2d 6209 . . 3 (𝜑 → (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
20 upgrun.un . . . 4 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
2120dmeqd 5463 . . . . 5 (𝜑 → dom (iEdg‘𝑈) = dom (𝐸𝐹))
22 dmun 5468 . . . . 5 dom (𝐸𝐹) = (dom 𝐸 ∪ dom 𝐹)
2321, 22syl6eq 2821 . . . 4 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
24 upgrun.v . . . . . . 7 (𝜑 → (Vtx‘𝑈) = 𝑉)
2524pweqd 4303 . . . . . 6 (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
2625difeq1d 3878 . . . . 5 (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2726rabeqdv 3344 . . . 4 (𝜑 → {𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
2820, 23, 27feq123d 6173 . . 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 247 . 2 (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶{𝑥 ∈ (𝒫 (Vtx‘𝑈) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
30 upgrun.u . . 3 (𝜑𝑈𝑊)
31 eqid 2771 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
32 eqid 2771 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
3331, 32isupgr 26200 . . 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 247 1 (𝜑𝑈 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {crab 3065  cdif 3720  cun 3721  cin 3722  c0 4063  𝒫 cpw 4298  {csn 4317   class class class wbr 4787  dom cdm 5250  wf 6026  cfv 6030  cle 10281  2c2 11276  chash 13321  Vtxcvtx 26095  iEdgciedg 26096  UPGraphcupgr 26196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-upgr 26198
This theorem is referenced by:  upgrunop  26235  uspgrun  26302
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