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Theorem uhgrunop 26423
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 5164 . . 3 𝑉, (𝐸𝐹)⟩ ∈ V
98a1i 11 . 2 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ V)
105fvexi 6460 . . . 4 𝑉 ∈ V
113fvexi 6460 . . . . 5 𝐸 ∈ V
124fvexi 6460 . . . . 5 𝐹 ∈ V
1311, 12unex 7233 . . . 4 (𝐸𝐹) ∈ V
1410, 13pm3.2i 464 . . 3 (𝑉 ∈ V ∧ (𝐸𝐹) ∈ V)
15 opvtxfv 26352 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
1614, 15mp1i 13 . 2 (𝜑 → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
17 opiedgfv 26355 . . 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 26422 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  cun 3790  cin 3791  c0 4141  cop 4404  dom cdm 5355  cfv 6135  Vtxcvtx 26344  iEdgciedg 26345  UHGraphcuhgr 26404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-1st 7445  df-2nd 7446  df-vtx 26346  df-iedg 26347  df-uhgr 26406
This theorem is referenced by:  ushgrunop  26425
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