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Theorem upgrunop 27719
Description: The union of two pseudographs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are pseudographs, then 𝑉, 𝐸𝐹 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)
Hypotheses
Ref Expression
upgrun.g (𝜑𝐺 ∈ UPGraph)
upgrun.h (𝜑𝐻 ∈ UPGraph)
upgrun.e 𝐸 = (iEdg‘𝐺)
upgrun.f 𝐹 = (iEdg‘𝐻)
upgrun.vg 𝑉 = (Vtx‘𝐺)
upgrun.vh (𝜑 → (Vtx‘𝐻) = 𝑉)
upgrun.i (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
Assertion
Ref Expression
upgrunop (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph)

Proof of Theorem upgrunop
StepHypRef Expression
1 upgrun.g . 2 (𝜑𝐺 ∈ UPGraph)
2 upgrun.h . 2 (𝜑𝐻 ∈ UPGraph)
3 upgrun.e . 2 𝐸 = (iEdg‘𝐺)
4 upgrun.f . 2 𝐹 = (iEdg‘𝐻)
5 upgrun.vg . 2 𝑉 = (Vtx‘𝐺)
6 upgrun.vh . 2 (𝜑 → (Vtx‘𝐻) = 𝑉)
7 upgrun.i . 2 (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
8 opex 5403 . . 3 𝑉, (𝐸𝐹)⟩ ∈ V
98a1i 11 . 2 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ V)
105fvexi 6833 . . . 4 𝑉 ∈ V
113fvexi 6833 . . . . 5 𝐸 ∈ V
124fvexi 6833 . . . . 5 𝐹 ∈ V
1311, 12unex 7650 . . . 4 (𝐸𝐹) ∈ V
1410, 13pm3.2i 471 . . 3 (𝑉 ∈ V ∧ (𝐸𝐹) ∈ V)
15 opvtxfv 27604 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
1614, 15mp1i 13 . 2 (𝜑 → (Vtx‘⟨𝑉, (𝐸𝐹)⟩) = 𝑉)
17 opiedgfv 27607 . . 3 ((𝑉 ∈ V ∧ (𝐸𝐹) ∈ V) → (iEdg‘⟨𝑉, (𝐸𝐹)⟩) = (𝐸𝐹))
1814, 17mp1i 13 . 2 (𝜑 → (iEdg‘⟨𝑉, (𝐸𝐹)⟩) = (𝐸𝐹))
191, 2, 3, 4, 5, 6, 7, 9, 16, 18upgrun 27718 1 (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  cun 3895  cin 3896  c0 4268  cop 4578  dom cdm 5614  cfv 6473  Vtxcvtx 27596  iEdgciedg 27597  UPGraphcupgr 27680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-fv 6481  df-1st 7891  df-2nd 7892  df-vtx 27598  df-iedg 27599  df-upgr 27682
This theorem is referenced by:  uspgrunop  27786
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