Theorem ushgrun 25742

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 25735	. . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph )
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph )
4		ushgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph )
5		ushgruhgr 25735	. . 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 25740	1 ⊢ (𝜑 → 𝑈 ∈ UHGraph )

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  dom cdm 5038  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722   USHGraph cushgr 25723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812  df-uhgr 25724  df-ushgr 25725

This theorem is referenced by: (None)