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Theorem umgr2v2evtxel 27870
Description: A vertex in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
Hypothesis
Ref Expression
umgr2v2evtx.g 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩
Assertion
Ref Expression
umgr2v2evtxel ((𝑉𝑊𝐴𝑉) → 𝐴 ∈ (Vtx‘𝐺))

Proof of Theorem umgr2v2evtxel
StepHypRef Expression
1 umgr2v2evtx.g . . 3 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩
21umgr2v2evtx 27869 . 2 (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)
3 eqcom 2746 . . . . 5 ((Vtx‘𝐺) = 𝑉𝑉 = (Vtx‘𝐺))
43biimpi 215 . . . 4 ((Vtx‘𝐺) = 𝑉𝑉 = (Vtx‘𝐺))
54eleq2d 2825 . . 3 ((Vtx‘𝐺) = 𝑉 → (𝐴𝑉𝐴 ∈ (Vtx‘𝐺)))
65biimpcd 248 . 2 (𝐴𝑉 → ((Vtx‘𝐺) = 𝑉𝐴 ∈ (Vtx‘𝐺)))
72, 6mpan9 506 1 ((𝑉𝑊𝐴𝑉) → 𝐴 ∈ (Vtx‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1541  wcel 2109  {cpr 4568  cop 4572  cfv 6430  0cc0 10855  1c1 10856  Vtxcvtx 27347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-iota 6388  df-fun 6432  df-fv 6438  df-1st 7817  df-vtx 27349
This theorem is referenced by:  umgr2v2enb1  27874  umgr2v2evd2  27875
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