Theorem ushgrun 29113

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 (𝐸 ∪ 𝐹) of the (indexed) edges. (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) = ∅)
ushgrun.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
ushgrun.v	⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
ushgrun.un	⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))

Assertion

Ref	Expression
ushgrun	⊢ (𝜑 → 𝑈 ∈ UHGraph)

Proof of Theorem ushgrun

Step	Hyp	Ref	Expression
1		ushgrun.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ USHGraph)
2		ushgruhgr 29106	. . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
4		ushgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph)
5		ushgruhgr 29106	. . 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		ushgrun.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
13		ushgrun.v	. 2 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
14		ushgrun.un	. 2 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹))
15	3, 6, 7, 8, 9, 10, 11, 12, 13, 14	uhgrun 29111	1 ⊢ (𝜑 → 𝑈 ∈ UHGraph)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   ∩ cin 3975  ∅c0 4352  dom cdm 5700  ‘cfv 6575  Vtxcvtx 29033  iEdgciedg 29034  UHGraphcuhgr 29093  USHGraphcushgr 29094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fv 6583  df-uhgr 29095  df-ushgr 29096

This theorem is referenced by: (None)