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

Theorem uspgrloopvtxel 27281
Description: A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27014). (Contributed by AV, 17-Dec-2020.)
Hypothesis
Ref Expression
uspgrloopvtx.g 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
Assertion
Ref Expression
uspgrloopvtxel ((𝑉𝑊𝑁𝑉) → 𝑁 ∈ (Vtx‘𝐺))

Proof of Theorem uspgrloopvtxel
StepHypRef Expression
1 uspgrloopvtx.g . . 3 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
21uspgrloopvtx 27280 . 2 (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)
3 eleq2 2899 . . . . 5 (𝑉 = (Vtx‘𝐺) → (𝑁𝑉𝑁 ∈ (Vtx‘𝐺)))
43biimpd 231 . . . 4 (𝑉 = (Vtx‘𝐺) → (𝑁𝑉𝑁 ∈ (Vtx‘𝐺)))
54eqcoms 2828 . . 3 ((Vtx‘𝐺) = 𝑉 → (𝑁𝑉𝑁 ∈ (Vtx‘𝐺)))
65com12 32 . 2 (𝑁𝑉 → ((Vtx‘𝐺) = 𝑉𝑁 ∈ (Vtx‘𝐺)))
72, 6mpan9 509 1 ((𝑉𝑊𝑁𝑉) → 𝑁 ∈ (Vtx‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  {csn 4539  cop 4545  cfv 6327  Vtxcvtx 26764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6286  df-fun 6329  df-fv 6335  df-1st 7663  df-vtx 26766
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator