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Theorem ushgrun 26424
Description: The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (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) = ∅)
ushgrun.u (𝜑𝑈𝑊)
ushgrun.v (𝜑 → (Vtx‘𝑈) = 𝑉)
ushgrun.un (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
Assertion
Ref Expression
ushgrun (𝜑𝑈 ∈ UHGraph)

Proof of Theorem ushgrun
StepHypRef Expression
1 ushgrun.g . . 3 (𝜑𝐺 ∈ USHGraph)
2 ushgruhgr 26417 . . 3 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
31, 2syl 17 . 2 (𝜑𝐺 ∈ UHGraph)
4 ushgrun.h . . 3 (𝜑𝐻 ∈ USHGraph)
5 ushgruhgr 26417 . . 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) = ∅)
12 ushgrun.u . 2 (𝜑𝑈𝑊)
13 ushgrun.v . 2 (𝜑 → (Vtx‘𝑈) = 𝑉)
14 ushgrun.un . 2 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
153, 6, 7, 8, 9, 10, 11, 12, 13, 14uhgrun 26422 1 (𝜑𝑈 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  cun 3790  cin 3791  c0 4141  dom cdm 5355  cfv 6135  Vtxcvtx 26344  iEdgciedg 26345  UHGraphcuhgr 26404  USHGraphcushgr 26405
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-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
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-id 5261  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-f1 6140  df-fv 6143  df-uhgr 26406  df-ushgr 26407
This theorem is referenced by: (None)
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