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Theorem uhgrun 28599
Description: The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
Hypotheses
Ref Expression
uhgrun.g (𝜑𝐺 ∈ UHGraph)
uhgrun.h (𝜑𝐻 ∈ UHGraph)
uhgrun.e 𝐸 = (iEdg‘𝐺)
uhgrun.f 𝐹 = (iEdg‘𝐻)
uhgrun.vg 𝑉 = (Vtx‘𝐺)
uhgrun.vh (𝜑 → (Vtx‘𝐻) = 𝑉)
uhgrun.i (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
uhgrun.u (𝜑𝑈𝑊)
uhgrun.v (𝜑 → (Vtx‘𝑈) = 𝑉)
uhgrun.un (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
Assertion
Ref Expression
uhgrun (𝜑𝑈 ∈ UHGraph)

Proof of Theorem uhgrun
StepHypRef Expression
1 uhgrun.g . . . . 5 (𝜑𝐺 ∈ UHGraph)
2 uhgrun.vg . . . . . 6 𝑉 = (Vtx‘𝐺)
3 uhgrun.e . . . . . 6 𝐸 = (iEdg‘𝐺)
42, 3uhgrf 28587 . . . . 5 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
51, 4syl 17 . . . 4 (𝜑𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
6 uhgrun.h . . . . . 6 (𝜑𝐻 ∈ UHGraph)
7 eqid 2730 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘𝐻)
8 uhgrun.f . . . . . . 7 𝐹 = (iEdg‘𝐻)
97, 8uhgrf 28587 . . . . . 6 (𝐻 ∈ UHGraph → 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))
106, 9syl 17 . . . . 5 (𝜑𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅}))
11 uhgrun.vh . . . . . . . . 9 (𝜑 → (Vtx‘𝐻) = 𝑉)
1211eqcomd 2736 . . . . . . . 8 (𝜑𝑉 = (Vtx‘𝐻))
1312pweqd 4620 . . . . . . 7 (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
1413difeq1d 4122 . . . . . 6 (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
1514feq3d 6705 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅})))
1610, 15mpbird 256 . . . 4 (𝜑𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}))
17 uhgrun.i . . . 4 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
185, 16, 17fun2d 6756 . . 3 (𝜑 → (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅}))
19 uhgrun.un . . . 4 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
2019dmeqd 5906 . . . . 5 (𝜑 → dom (iEdg‘𝑈) = dom (𝐸𝐹))
21 dmun 5911 . . . . 5 dom (𝐸𝐹) = (dom 𝐸 ∪ dom 𝐹)
2220, 21eqtrdi 2786 . . . 4 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
23 uhgrun.v . . . . . 6 (𝜑 → (Vtx‘𝑈) = 𝑉)
2423pweqd 4620 . . . . 5 (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
2524difeq1d 4122 . . . 4 (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2619, 22, 25feq123d 6707 . . 3 (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}) ↔ (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅})))
2718, 26mpbird 256 . 2 (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}))
28 uhgrun.u . . 3 (𝜑𝑈𝑊)
29 eqid 2730 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
30 eqid 2730 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
3129, 30isuhgr 28585 . . 3 (𝑈𝑊 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})))
3228, 31syl 17 . 2 (𝜑 → (𝑈 ∈ UHGraph ↔ (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅})))
3327, 32mpbird 256 1 (𝜑𝑈 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2104  cdif 3946  cun 3947  cin 3948  c0 4323  𝒫 cpw 4603  {csn 4629  dom cdm 5677  wf 6540  cfv 6544  Vtxcvtx 28521  iEdgciedg 28522  UHGraphcuhgr 28581
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-uhgr 28583
This theorem is referenced by:  uhgrunop  28600  ushgrun  28601
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