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Theorem ushgrunop 26873
 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, 𝐸 ∪ 𝐹⟩ 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 26865 . . 3 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
31, 2syl 17 . 2 (𝜑𝐺 ∈ UHGraph)
4 ushgrun.h . . 3 (𝜑𝐻 ∈ USHGraph)
5 ushgruhgr 26865 . . 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 26871 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112   ∪ cun 3882   ∩ cin 3883  ∅c0 4246  ⟨cop 4534  dom cdm 5523  ‘cfv 6328  Vtxcvtx 26792  iEdgciedg 26793  UHGraphcuhgr 26852  USHGraphcushgr 26853 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336  df-1st 7675  df-2nd 7676  df-vtx 26794  df-iedg 26795  df-uhgr 26854  df-ushgr 26855 This theorem is referenced by: (None)
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