Theorem uspgrunop 29276

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 29265	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
4		uspgrun.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph)
5		uspgrupgr 29265	. . 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 29206	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∪ cun 3888   ∩ cin 3889  ∅c0 4274  ⟨cop 4574  dom cdm 5626  ‘cfv 6494  Vtxcvtx 29083  iEdgciedg 29084  UPGraphcupgr 29167  USPGraphcuspgr 29235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fv 6502  df-1st 7937  df-2nd 7938  df-vtx 29085  df-iedg 29086  df-upgr 29169  df-uspgr 29237

This theorem is referenced by: (None)