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Theorem umgr2v2evtx 28511
Description: The set of vertices 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, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩}⟩
Assertion
Ref Expression
umgr2v2evtx (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)

Proof of Theorem umgr2v2evtx
StepHypRef Expression
1 umgr2v2evtx.g . . 3 𝐺 = ⟚𝑉, {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩}⟩
21fveq2i 6846 . 2 (Vtx‘𝐺) = (Vtx‘⟚𝑉, {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩}⟩)
3 prex 5390 . . 3 {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩} ∈ V
4 opvtxfv 27997 . . 3 ((𝑉 ∈ 𝑊 ∧ {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩} ∈ V) → (Vtx‘⟚𝑉, {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩}⟩) = 𝑉)
53, 4mpan2 690 . 2 (𝑉 ∈ 𝑊 → (Vtx‘⟚𝑉, {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩}⟩) = 𝑉)
62, 5eqtrid 2785 1 (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3444  {cpr 4589  âŸšcop 4593  â€˜cfv 6497  0cc0 11056  1c1 11057  Vtxcvtx 27989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-vtx 27991
This theorem is referenced by:  umgr2v2evtxel  28512  umgr2v2e  28515  umgr2v2enb1  28516
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