MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  umgr2v2evtxel Structured version   Visualization version   GIF version

Theorem umgr2v2evtxel 29555
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 29554 . 2 (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)
3 eqcom 2742 . . . . 5 ((Vtx‘𝐺) = 𝑉𝑉 = (Vtx‘𝐺))
43biimpi 216 . . . 4 ((Vtx‘𝐺) = 𝑉𝑉 = (Vtx‘𝐺))
54eleq2d 2825 . . 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 1537  wcel 2106  {cpr 4633  cop 4637  cfv 6563  0cc0 11153  1c1 11154  Vtxcvtx 29028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-vtx 29030
This theorem is referenced by:  umgr2v2enb1  29559  umgr2v2evd2  29560
  Copyright terms: Public domain W3C validator