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Theorem 1loopgredg 40716
 Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
Hypotheses
Ref Expression
1loopgruspgr.v (𝜑 → (Vtx‘𝐺) = 𝑉)
1loopgruspgr.a (𝜑𝐴𝑋)
1loopgruspgr.n (𝜑𝑁𝑉)
1loopgruspgr.i (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})
Assertion
Ref Expression
1loopgredg (𝜑 → (Edg‘𝐺) = {{𝑁}})

Proof of Theorem 1loopgredg
StepHypRef Expression
1 1loopgruspgr.n . . . . 5 (𝜑𝑁𝑉)
2 1loopgruspgr.v . . . . 5 (𝜑 → (Vtx‘𝐺) = 𝑉)
31, 2eleqtrrd 2691 . . . 4 (𝜑𝑁 ∈ (Vtx‘𝐺))
43elfvexd 6132 . . 3 (𝜑𝐺 ∈ V)
5 edgaval 25794 . . 3 (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
64, 5syl 17 . 2 (𝜑 → (Edg‘𝐺) = ran (iEdg‘𝐺))
7 1loopgruspgr.i . . 3 (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})
87rneqd 5274 . 2 (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
9 1loopgruspgr.a . . 3 (𝜑𝐴𝑋)
10 rnsnopg 5532 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
119, 10syl 17 . 2 (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
126, 8, 113eqtrd 2648 1 (𝜑 → (Edg‘𝐺) = {{𝑁}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  ran crn 5039  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-edga 25793 This theorem is referenced by:  1loopgrnb0  40717  1loopgrvd2  40718
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