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Theorem uspgrun 29021
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 29011 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
31, 2syl 17 . 2 (𝜑𝐺 ∈ UPGraph)
4 uspgrun.h . . 3 (𝜑𝐻 ∈ USPGraph)
5 uspgrupgr 29011 . . 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 28951 1 (𝜑𝑈 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cun 3947  cin 3948  c0 4326  dom cdm 5682  cfv 6553  Vtxcvtx 28829  iEdgciedg 28830  UPGraphcupgr 28913  USPGraphcuspgr 28981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fv 6561  df-upgr 28915  df-uspgr 28983
This theorem is referenced by: (None)
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