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Theorem uhgrun 28842
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 28830 . . . . 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 2726 . . . . . . 7 (Vtx‘𝐻) = (Vtx‘𝐻)
8 uhgrun.f . . . . . . 7 𝐹 = (iEdg‘𝐻)
97, 8uhgrf 28830 . . . . . 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 2732 . . . . . . . 8 (𝜑𝑉 = (Vtx‘𝐻))
1312pweqd 4614 . . . . . . 7 (𝜑 → 𝒫 𝑉 = 𝒫 (Vtx‘𝐻))
1413difeq1d 4116 . . . . . 6 (𝜑 → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐻) ∖ {∅}))
1514feq3d 6698 . . . . 5 (𝜑 → (𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (Vtx‘𝐻) ∖ {∅})))
1610, 15mpbird 257 . . . 4 (𝜑𝐹:dom 𝐹⟶(𝒫 𝑉 ∖ {∅}))
17 uhgrun.i . . . 4 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
185, 16, 17fun2d 6749 . . 3 (𝜑 → (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅}))
19 uhgrun.un . . . 4 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
2019dmeqd 5899 . . . . 5 (𝜑 → dom (iEdg‘𝑈) = dom (𝐸𝐹))
21 dmun 5904 . . . . 5 dom (𝐸𝐹) = (dom 𝐸 ∪ dom 𝐹)
2220, 21eqtrdi 2782 . . . 4 (𝜑 → dom (iEdg‘𝑈) = (dom 𝐸 ∪ dom 𝐹))
23 uhgrun.v . . . . . 6 (𝜑 → (Vtx‘𝑈) = 𝑉)
2423pweqd 4614 . . . . 5 (𝜑 → 𝒫 (Vtx‘𝑈) = 𝒫 𝑉)
2524difeq1d 4116 . . . 4 (𝜑 → (𝒫 (Vtx‘𝑈) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2619, 22, 25feq123d 6700 . . 3 (𝜑 → ((iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}) ↔ (𝐸𝐹):(dom 𝐸 ∪ dom 𝐹)⟶(𝒫 𝑉 ∖ {∅})))
2718, 26mpbird 257 . 2 (𝜑 → (iEdg‘𝑈):dom (iEdg‘𝑈)⟶(𝒫 (Vtx‘𝑈) ∖ {∅}))
28 uhgrun.u . . 3 (𝜑𝑈𝑊)
29 eqid 2726 . . . 4 (Vtx‘𝑈) = (Vtx‘𝑈)
30 eqid 2726 . . . 4 (iEdg‘𝑈) = (iEdg‘𝑈)
3129, 30isuhgr 28828 . . 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 257 1 (𝜑𝑈 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  cdif 3940  cun 3941  cin 3942  c0 4317  𝒫 cpw 4597  {csn 4623  dom cdm 5669  wf 6533  cfv 6537  Vtxcvtx 28764  iEdgciedg 28765  UHGraphcuhgr 28824
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-uhgr 28826
This theorem is referenced by:  uhgrunop  28843  ushgrun  28844
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