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Theorem uspgrun 40415
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 40406 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
31, 2syl 17 . 2 (𝜑𝐺 ∈ UPGraph )
4 uspgrun.h . . 3 (𝜑𝐻 ∈ USPGraph )
5 uspgrupgr 40406 . . 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 25784 1 (𝜑𝑈 ∈ UPGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  cun 3538  cin 3539  c0 3874  dom cdm 5038  cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747   USPGraph cuspgr 40378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812  df-upgr 25749  df-uspgr 40380
This theorem is referenced by: (None)
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