Theorem ushgrun 29711

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 set 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 29704	. . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
4		ushgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph)
5		ushgruhgr 29704	. . 3 ⊢ (𝐻 ∈ USHGraph → 𝐻 ∈ UHGraph)
6	4, 5	syl 18	. 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 29709	1 ⊢ (𝜑 → 𝑈 ∈ UHGraph)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  dom cdm 5658  ‘cfv 6566  Vtxcvtx 29631  iEdgciedg 29632  UHGraphcuhgr 29691  USHGraphcushgr 29692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fv 6574  df-uhgr 29693  df-ushgr 29694

This theorem is used by: (None)