Theorem uhgrunop 29048

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 5399	. . 3 ⊢ ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V
9	8	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V)
10	5	fvexi 6831	. . . 4 ⊢ 𝑉 ∈ V
11	3	fvexi 6831	. . . . 5 ⊢ 𝐸 ∈ V
12	4	fvexi 6831	. . . . 5 ⊢ 𝐹 ∈ V
13	11, 12	unex 7672	. . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V
14	10, 13	pm3.2i 470	. . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V)
15		opvtxfv 28977	. . 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 28980	. . 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 29047	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∪ cun 3895   ∩ cin 3896  ∅c0 4278  ⟨cop 4577  dom cdm 5611  ‘cfv 6476  Vtxcvtx 28969  iEdgciedg 28970  UHGraphcuhgr 29029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-1st 7916  df-2nd 7917  df-vtx 28971  df-iedg 28972  df-uhgr 29031

This theorem is referenced by:  ushgrunop  29050