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Theorem uspgrun 29206
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 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) = ∅)
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 29136 1 (𝜑𝑈 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cun 3948  cin 3949  c0 4332  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-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
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-clab 2714  df-cleq 2728  df-clel 2815  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-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-upgr 29100  df-uspgr 29168
This theorem is referenced by: (None)
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