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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 (𝐸 ∪ 𝐹) 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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