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Theorem umgr2v2evtxel 29540
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 29539 . 2 (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)
3 eqcom 2744 . . . . 5 ((Vtx‘𝐺) = 𝑉𝑉 = (Vtx‘𝐺))
43biimpi 216 . . . 4 ((Vtx‘𝐺) = 𝑉𝑉 = (Vtx‘𝐺))
54eleq2d 2827 . . 3 ((Vtx‘𝐺) = 𝑉 → (𝐴𝑉𝐴 ∈ (Vtx‘𝐺)))
65biimpcd 249 . 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 1540  wcel 2108  {cpr 4628  cop 4632  cfv 6561  0cc0 11155  1c1 11156  Vtxcvtx 29013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-vtx 29015
This theorem is referenced by:  umgr2v2enb1  29544  umgr2v2evd2  29545
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