Theorem ushgrun 29056

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 29049	. . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
4		ushgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph)
5		ushgruhgr 29049	. . 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 29054	1 ⊢ (𝜑 → 𝑈 ∈ UHGraph)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ∪ cun 3896   ∩ cin 3897  ∅c0 4282  dom cdm 5619  ‘cfv 6486  Vtxcvtx 28976  iEdgciedg 28977  UHGraphcuhgr 29036  USHGraphcushgr 29037

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 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

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-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fv 6494  df-uhgr 29038  df-ushgr 29039

This theorem is referenced by: (None)