Theorem uhgrunop 29019

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

Proof of Theorem uhgrunop

Step	Hyp	Ref	Expression
1		uhgrun.g	. 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
2		uhgrun.h	. 2 ⊢ (𝜑 → 𝐻 ∈ 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 5449	. . 3 ⊢ ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V
9	8	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V)
10	5	fvexi 6899	. . . 4 ⊢ 𝑉 ∈ V
11	3	fvexi 6899	. . . . 5 ⊢ 𝐸 ∈ V
12	4	fvexi 6899	. . . . 5 ⊢ 𝐹 ∈ V
13	11, 12	unex 7745	. . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V
14	10, 13	pm3.2i 470	. . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V)
15		opvtxfv 28948	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = 𝑉)
16	14, 15	mp1i 13	. 2 ⊢ (𝜑 → (Vtx‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = 𝑉)
17		opiedgfv 28951	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (iEdg‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = (𝐸 ∪ 𝐹))
18	14, 17	mp1i 13	. 2 ⊢ (𝜑 → (iEdg‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = (𝐸 ∪ 𝐹))
19	1, 2, 3, 4, 5, 6, 7, 9, 16, 18	uhgrun 29018	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463   ∪ cun 3929   ∩ cin 3930  ∅c0 4313  ⟨cop 4612  dom cdm 5665  ‘cfv 6540  Vtxcvtx 28940  iEdgciedg 28941  UHGraphcuhgr 29000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-1st 7995  df-2nd 7996  df-vtx 28942  df-iedg 28943  df-uhgr 29002

This theorem is referenced by:  ushgrunop  29021