Theorem ushgrunop 27447

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	⊢ (𝜑 → 𝐺 ∈ USHGraph)
ushgrun.h	⊢ (𝜑 → 𝐻 ∈ 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, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph)

Proof of Theorem ushgrunop

Step	Hyp	Ref	Expression
1		ushgrun.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ USHGraph)
2		ushgruhgr 27439	. . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
4		ushgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph)
5		ushgruhgr 27439	. . 3 ⊢ (𝐻 ∈ USHGraph → 𝐻 ∈ UHGraph)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐻 ∈ 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	3, 6, 7, 8, 9, 10, 11	uhgrunop 27445	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ∩ cin 3886  ∅c0 4256  ⟨cop 4567  dom cdm 5589  ‘cfv 6433  Vtxcvtx 27366  iEdgciedg 27367  UHGraphcuhgr 27426  USHGraphcushgr 27427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441  df-1st 7831  df-2nd 7832  df-vtx 27368  df-iedg 27369  df-uhgr 27428  df-ushgr 27429

This theorem is referenced by: (None)