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Theorem uspgrun 27300
Description: The union 𝑈 of two simple pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)
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) = ∅)
uspgrun.u (𝜑𝑈𝑊)
uspgrun.v (𝜑 → (Vtx‘𝑈) = 𝑉)
uspgrun.un (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
Assertion
Ref Expression
uspgrun (𝜑𝑈 ∈ UPGraph)

Proof of Theorem uspgrun
StepHypRef Expression
1 uspgrun.g . . 3 (𝜑𝐺 ∈ USPGraph)
2 uspgrupgr 27291 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
31, 2syl 17 . 2 (𝜑𝐺 ∈ UPGraph)
4 uspgrun.h . . 3 (𝜑𝐻 ∈ USPGraph)
5 uspgrupgr 27291 . . 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) = ∅)
12 uspgrun.u . 2 (𝜑𝑈𝑊)
13 uspgrun.v . 2 (𝜑 → (Vtx‘𝑈) = 𝑉)
14 uspgrun.un . 2 (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))
153, 6, 7, 8, 9, 10, 11, 12, 13, 14upgrun 27233 1 (𝜑𝑈 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2111  cun 3879  cin 3880  c0 4252  dom cdm 5566  cfv 6398  Vtxcvtx 27111  iEdgciedg 27112  UPGraphcupgr 27195  USPGraphcuspgr 27263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-sbc 3710  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-id 5470  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fv 6406  df-upgr 27197  df-uspgr 27265
This theorem is referenced by: (None)
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