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Theorem uspgrloopvtx 29316
Description: The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29049). (Contributed by AV, 17-Dec-2020.)
Hypothesis
Ref Expression
uspgrloopvtx.g 𝐺 = ⟚𝑉, {⟚𝐎, {𝑁}⟩}⟩
Assertion
Ref Expression
uspgrloopvtx (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)

Proof of Theorem uspgrloopvtx
StepHypRef Expression
1 uspgrloopvtx.g . . 3 𝐺 = ⟚𝑉, {⟚𝐎, {𝑁}⟩}⟩
21fveq2i 6894 . 2 (Vtx‘𝐺) = (Vtx‘⟚𝑉, {⟚𝐎, {𝑁}⟩}⟩)
3 snex 5427 . . 3 {⟚𝐎, {𝑁}⟩} ∈ V
4 opvtxfv 28804 . . 3 ((𝑉 ∈ 𝑊 ∧ {⟚𝐎, {𝑁}⟩} ∈ V) → (Vtx‘⟚𝑉, {⟚𝐎, {𝑁}⟩}⟩) = 𝑉)
53, 4mpan2 690 . 2 (𝑉 ∈ 𝑊 → (Vtx‘⟚𝑉, {⟚𝐎, {𝑁}⟩}⟩) = 𝑉)
62, 5eqtrid 2779 1 (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3469  {csn 4624  âŸšcop 4630  â€˜cfv 6542  Vtxcvtx 28796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  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 7987  df-vtx 28798
This theorem is referenced by:  uspgrloopvtxel  29317  uspgrloopnb0  29320  uspgrloopvd2  29321
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