Theorem uspgrunop 29280

Description: The union of two simple pseudographs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are simple pseudographs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
uspgrun.g	⊢ (𝜑 → 𝐺 ∈ USPGraph)
uspgrun.h	⊢ (𝜑 → 𝐻 ∈ USPGraph)
uspgrun.e	⊢ 𝐸 = (iEdg‘𝐺)
uspgrun.f	⊢ 𝐹 = (iEdg‘𝐻)
uspgrun.vg	⊢ 𝑉 = (Vtx‘𝐺)
uspgrun.vh	⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
uspgrun.i	⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)

Assertion

Ref	Expression
uspgrunop	⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

Proof of Theorem uspgrunop

Step	Hyp	Ref	Expression
1		uspgrun.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ USPGraph)
2		uspgrupgr 29269	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
4		uspgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph)
5		uspgrupgr 29269	. . 3 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐻 ∈ UPGraph)
7		uspgrun.e	. 2 ⊢ 𝐸 = (iEdg‘𝐺)
8		uspgrun.f	. 2 ⊢ 𝐹 = (iEdg‘𝐻)
9		uspgrun.vg	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
10		uspgrun.vh	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
11		uspgrun.i	. 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
12	3, 6, 7, 8, 9, 10, 11	upgrunop 29210	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121   ∪ cun 3883   ∩ cin 3884  ∅c0 4264  ⟨cop 4564  dom cdm 5621  ‘cfv 6489  Vtxcvtx 29087  iEdgciedg 29088  UPGraphcupgr 29171  USPGraphcuspgr 29239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fv 6497  df-1st 7935  df-2nd 7936  df-vtx 29089  df-iedg 29090  df-upgr 29173  df-uspgr 29241

This theorem is referenced by: (None)