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Theorem uhgrunop 26191
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 5061 . . 3 𝑉, (𝐸𝐹)⟩ ∈ V
98a1i 11 . 2 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ V)
105fvexi 6345 . . . 4 𝑉 ∈ V
113fvexi 6345 . . . . 5 𝐸 ∈ V
124fvexi 6345 . . . . 5 𝐹 ∈ V
1311, 12unex 7107 . . . 4 (𝐸𝐹) ∈ V
1410, 13pm3.2i 456 . . 3 (𝑉 ∈ V ∧ (𝐸𝐹) ∈ V)
15 opvtxfv 26105 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
1614, 15mp1i 13 . 2 (𝜑 → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
17 opiedgfv 26108 . . 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 26190 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  cin 3722  c0 4063  cop 4323  dom cdm 5250  cfv 6030  Vtxcvtx 26095  iEdgciedg 26096  UHGraphcuhgr 26172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-1st 7319  df-2nd 7320  df-vtx 26097  df-iedg 26098  df-uhgr 26174
This theorem is referenced by:  ushgrunop  26193
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