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Theorem ushgrun 28840
Description: The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.)
Hypotheses
Ref Expression
ushgrun.g (𝜑𝐺 ∈ USHGraph)
ushgrun.h (𝜑𝐻 ∈ USHGraph)
ushgrun.e 𝐸 = (iEdg‘𝐺)
ushgrun.f 𝐹 = (iEdg‘𝐻)
ushgrun.vg 𝑉 = (Vtx‘𝐺)
ushgrun.vh (𝜑 → (Vtx‘𝐻) = 𝑉)
ushgrun.i (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
ushgrun.u (𝜑𝑈𝑊)
ushgrun.v (𝜑 → (Vtx‘𝑈) = 𝑉)
ushgrun.un (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
Assertion
Ref Expression
ushgrun (𝜑𝑈 ∈ UHGraph)

Proof of Theorem ushgrun
StepHypRef Expression
1 ushgrun.g . . 3 (𝜑𝐺 ∈ USHGraph)
2 ushgruhgr 28833 . . 3 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
31, 2syl 17 . 2 (𝜑𝐺 ∈ UHGraph)
4 ushgrun.h . . 3 (𝜑𝐻 ∈ USHGraph)
5 ushgruhgr 28833 . . 3 (𝐻 ∈ USHGraph → 𝐻 ∈ UHGraph)
64, 5syl 17 . 2 (𝜑𝐻 ∈ UHGraph)
7 ushgrun.e . 2 𝐸 = (iEdg‘𝐺)
8 ushgrun.f . 2 𝐹 = (iEdg‘𝐻)
9 ushgrun.vg . 2 𝑉 = (Vtx‘𝐺)
10 ushgrun.vh . 2 (𝜑 → (Vtx‘𝐻) = 𝑉)
11 ushgrun.i . 2 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
12 ushgrun.u . 2 (𝜑𝑈𝑊)
13 ushgrun.v . 2 (𝜑 → (Vtx‘𝑈) = 𝑉)
14 ushgrun.un . 2 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
153, 6, 7, 8, 9, 10, 11, 12, 13, 14uhgrun 28838 1 (𝜑𝑈 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cun 3941  cin 3942  c0 4317  dom cdm 5669  cfv 6536  Vtxcvtx 28760  iEdgciedg 28761  UHGraphcuhgr 28820  USHGraphcushgr 28821
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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544  df-uhgr 28822  df-ushgr 28823
This theorem is referenced by: (None)
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