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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, 𝐸𝐹 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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