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Theorem umgr2v2evtx 29377
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 6894 . 2 (Vtx‘𝐺) = (Vtx‘⟚𝑉, {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩}⟩)
3 prex 5428 . . 3 {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩} ∈ V
4 opvtxfv 28859 . . 3 ((𝑉 ∈ 𝑊 ∧ {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩} ∈ V) → (Vtx‘⟚𝑉, {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩}⟩) = 𝑉)
53, 4mpan2 689 . 2 (𝑉 ∈ 𝑊 → (Vtx‘⟚𝑉, {⟹0, {𝐎, 𝐵}⟩, ⟹1, {𝐎, 𝐵}⟩}⟩) = 𝑉)
62, 5eqtrid 2777 1 (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3463  {cpr 4626  âŸšcop 4630  â€˜cfv 6542  0cc0 11136  1c1 11137  Vtxcvtx 28851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-1st 7989  df-vtx 28853
This theorem is referenced by:  umgr2v2evtxel  29378  umgr2v2e  29381  umgr2v2enb1  29382
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