MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uspgrunop Structured version   Visualization version   GIF version

Theorem uspgrunop 27131
Description: The union of two simple pseudographs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are simple pseudographs, then 𝑉, 𝐸𝐹 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.)
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) = ∅)
Assertion
Ref Expression
uspgrunop (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph)

Proof of Theorem uspgrunop
StepHypRef Expression
1 uspgrun.g . . 3 (𝜑𝐺 ∈ USPGraph)
2 uspgrupgr 27121 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
31, 2syl 17 . 2 (𝜑𝐺 ∈ UPGraph)
4 uspgrun.h . . 3 (𝜑𝐻 ∈ USPGraph)
5 uspgrupgr 27121 . . 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) = ∅)
123, 6, 7, 8, 9, 10, 11upgrunop 27064 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  cun 3841  cin 3842  c0 4211  cop 4522  dom cdm 5525  cfv 6339  Vtxcvtx 26941  iEdgciedg 26942  UPGraphcupgr 27025  USPGraphcuspgr 27093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fv 6347  df-1st 7714  df-2nd 7715  df-vtx 26943  df-iedg 26944  df-upgr 27027  df-uspgr 27095
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator