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Theorem uspgrunop 29207
Description: The union of two simple pseudographs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are simple pseudographs, then 𝑉, 𝐸𝐹 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident 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 (𝜑𝐺 ∈ USPGraph)
uspgrun.h (𝜑𝐻 ∈ 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, (𝐸𝐹)⟩ ∈ UPGraph)

Proof of Theorem uspgrunop
StepHypRef Expression
1 uspgrun.g . . 3 (𝜑𝐺 ∈ USPGraph)
2 uspgrupgr 29196 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
31, 2syl 17 . 2 (𝜑𝐺 ∈ UPGraph)
4 uspgrun.h . . 3 (𝜑𝐻 ∈ USPGraph)
5 uspgrupgr 29196 . . 3 (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
64, 5syl 17 . 2 (𝜑𝐻 ∈ 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) = ∅)
123, 6, 7, 8, 9, 10, 11upgrunop 29137 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cun 3948  cin 3949  c0 4332  cop 4631  dom cdm 5684  cfv 6560  Vtxcvtx 29014  iEdgciedg 29015  UPGraphcupgr 29098  USPGraphcuspgr 29166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fv 6568  df-1st 8015  df-2nd 8016  df-vtx 29016  df-iedg 29017  df-upgr 29100  df-uspgr 29168
This theorem is referenced by: (None)
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