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Theorem ushgrunop 26559
Description: The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If 𝑉, 𝐸 and 𝑉, 𝐹 are simple hypergraphs, then 𝑉, 𝐸𝐹 is a (not necessarily simple) hypergraph - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (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) = ∅)
Assertion
Ref Expression
ushgrunop (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)

Proof of Theorem ushgrunop
StepHypRef Expression
1 ushgrun.g . . 3 (𝜑𝐺 ∈ USHGraph)
2 ushgruhgr 26551 . . 3 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
31, 2syl 17 . 2 (𝜑𝐺 ∈ UHGraph)
4 ushgrun.h . . 3 (𝜑𝐻 ∈ USHGraph)
5 ushgruhgr 26551 . . 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) = ∅)
123, 6, 7, 8, 9, 10, 11uhgrunop 26557 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  cun 3821  cin 3822  c0 4172  cop 4441  dom cdm 5401  cfv 6182  Vtxcvtx 26478  iEdgciedg 26479  UHGraphcuhgr 26538  USHGraphcushgr 26539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fv 6190  df-1st 7495  df-2nd 7496  df-vtx 26480  df-iedg 26481  df-uhgr 26540  df-ushgr 26541
This theorem is referenced by: (None)
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