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Theorem upgrunop 25909
 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 (𝜑𝐺 ∈ UPGraph )
upgrun.h (𝜑𝐻 ∈ 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, (𝐸𝐹)⟩ ∈ UPGraph )

Proof of Theorem upgrunop
StepHypRef Expression
1 upgrun.g . 2 (𝜑𝐺 ∈ UPGraph )
2 upgrun.h . 2 (𝜑𝐻 ∈ 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 4893 . . 3 𝑉, (𝐸𝐹)⟩ ∈ V
98a1i 11 . 2 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ V)
10 fvex 6158 . . . . 5 (Vtx‘𝐺) ∈ V
115, 10eqeltri 2694 . . . 4 𝑉 ∈ V
12 fvex 6158 . . . . . 6 (iEdg‘𝐺) ∈ V
133, 12eqeltri 2694 . . . . 5 𝐸 ∈ V
14 fvex 6158 . . . . . 6 (iEdg‘𝐻) ∈ V
154, 14eqeltri 2694 . . . . 5 𝐹 ∈ V
1613, 15unex 6909 . . . 4 (𝐸𝐹) ∈ V
1711, 16pm3.2i 471 . . 3 (𝑉 ∈ V ∧ (𝐸𝐹) ∈ V)
18 opvtxfv 25784 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
1917, 18mp1i 13 . 2 (𝜑 → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
20 opiedgfv 25787 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (iEdg‘⟨𝑉, (𝐸𝐹)⟩) = (𝐸𝐹))
2117, 20mp1i 13 . 2 (𝜑 → (iEdg‘⟨𝑉, (𝐸𝐹)⟩) = (𝐸𝐹))
221, 2, 3, 4, 5, 6, 7, 9, 19, 21upgrun 25908 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ∪ cun 3553   ∩ cin 3554  ∅c0 3891  ⟨cop 4154  dom cdm 5074  ‘cfv 5847  Vtxcvtx 25774  iEdgciedg 25775   UPGraph cupgr 25871 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-1st 7113  df-2nd 7114  df-vtx 25776  df-iedg 25777  df-upgr 25873 This theorem is referenced by:  uspgrunop  25974
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