Theorem upgrunop 26898

Description: The union of two pseudographs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are pseudographs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

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

Assertion

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

Proof of Theorem upgrunop

Step	Hyp	Ref	Expression
1		upgrun.g	. 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
2		upgrun.h	. 2 ⊢ (𝜑 → 𝐻 ∈ UPGraph)
3		upgrun.e	. 2 ⊢ 𝐸 = (iEdg‘𝐺)
4		upgrun.f	. 2 ⊢ 𝐹 = (iEdg‘𝐻)
5		upgrun.vg	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
6		upgrun.vh	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
7		upgrun.i	. 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
8		opex 5348	. . 3 ⊢ ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V
9	8	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V)
10	5	fvexi 6678	. . . 4 ⊢ 𝑉 ∈ V
11	3	fvexi 6678	. . . . 5 ⊢ 𝐸 ∈ V
12	4	fvexi 6678	. . . . 5 ⊢ 𝐹 ∈ V
13	11, 12	unex 7463	. . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V
14	10, 13	pm3.2i 473	. . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V)
15		opvtxfv 26783	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = 𝑉)
16	14, 15	mp1i 13	. 2 ⊢ (𝜑 → (Vtx‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = 𝑉)
17		opiedgfv 26786	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (iEdg‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = (𝐸 ∪ 𝐹))
18	14, 17	mp1i 13	. 2 ⊢ (𝜑 → (iEdg‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = (𝐸 ∪ 𝐹))
19	1, 2, 3, 4, 5, 6, 7, 9, 16, 18	upgrun 26897	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∪ cun 3933   ∩ cin 3934  ∅c0 4290  ⟨cop 4566  dom cdm 5549  ‘cfv 6349  Vtxcvtx 26775  iEdgciedg 26776  UPGraphcupgr 26859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-1st 7683  df-2nd 7684  df-vtx 26777  df-iedg 26778  df-upgr 26861

This theorem is referenced by:  uspgrunop  26965