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Theorem uhgrunop 26847
 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, 𝐸 ∪ 𝐹⟩ 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 5329 . . 3 𝑉, (𝐸𝐹)⟩ ∈ V
98a1i 11 . 2 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ V)
105fvexi 6657 . . . 4 𝑉 ∈ V
113fvexi 6657 . . . . 5 𝐸 ∈ V
124fvexi 6657 . . . . 5 𝐹 ∈ V
1311, 12unex 7444 . . . 4 (𝐸𝐹) ∈ V
1410, 13pm3.2i 474 . . 3 (𝑉 ∈ V ∧ (𝐸𝐹) ∈ V)
15 opvtxfv 26776 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
1614, 15mp1i 13 . 2 (𝜑 → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
17 opiedgfv 26779 . . 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 26846 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3471   ∪ cun 3908   ∩ cin 3909  ∅c0 4266  ⟨cop 4546  dom cdm 5528  ‘cfv 6328  Vtxcvtx 26768  iEdgciedg 26769  UHGraphcuhgr 26828 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-1st 7664  df-2nd 7665  df-vtx 26770  df-iedg 26771  df-uhgr 26830 This theorem is referenced by:  ushgrunop  26849
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