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Theorem uhgrunop 28200
Description: The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If 𝑉, 𝐸 and 𝑉, 𝐹 are hypergraphs, then 𝑉, 𝐸𝐹 is a 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 Alexander van der Vekens, 27-Dec-2017.) (Revised 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) = ∅)
Assertion
Ref Expression
uhgrunop (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)

Proof of Theorem uhgrunop
StepHypRef Expression
1 uhgrun.g . 2 (𝜑𝐺 ∈ UHGraph)
2 uhgrun.h . 2 (𝜑𝐻 ∈ UHGraph)
3 uhgrun.e . 2 𝐸 = (iEdg‘𝐺)
4 uhgrun.f . 2 𝐹 = (iEdg‘𝐻)
5 uhgrun.vg . 2 𝑉 = (Vtx‘𝐺)
6 uhgrun.vh . 2 (𝜑 → (Vtx‘𝐻) = 𝑉)
7 uhgrun.i . 2 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
8 opex 5457 . . 3 𝑉, (𝐸𝐹)⟩ ∈ V
98a1i 11 . 2 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ V)
105fvexi 6892 . . . 4 𝑉 ∈ V
113fvexi 6892 . . . . 5 𝐸 ∈ V
124fvexi 6892 . . . . 5 𝐹 ∈ V
1311, 12unex 7716 . . . 4 (𝐸𝐹) ∈ V
1410, 13pm3.2i 471 . . 3 (𝑉 ∈ V ∧ (𝐸𝐹) ∈ V)
15 opvtxfv 28129 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
1614, 15mp1i 13 . 2 (𝜑 → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
17 opiedgfv 28132 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (iEdg‘⟨𝑉, (𝐸𝐹)⟩) = (𝐸𝐹))
1814, 17mp1i 13 . 2 (𝜑 → (iEdg‘⟨𝑉, (𝐸𝐹)⟩) = (𝐸𝐹))
191, 2, 3, 4, 5, 6, 7, 9, 16, 18uhgrun 28199 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3473  cun 3942  cin 3943  c0 4318  cop 4628  dom cdm 5669  cfv 6532  Vtxcvtx 28121  iEdgciedg 28122  UHGraphcuhgr 28181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-1st 7957  df-2nd 7958  df-vtx 28123  df-iedg 28124  df-uhgr 28183
This theorem is referenced by:  ushgrunop  28202
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