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Theorem 1loopgredg 26453
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 edgval 25986 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . 2 (𝜑 → (Edg‘𝐺) = ran (iEdg‘𝐺))
3 1loopgruspgr.i . . 3 (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})
43rneqd 5385 . 2 (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
5 1loopgruspgr.a . . 3 (𝜑𝐴𝑋)
6 rnsnopg 5650 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
75, 6syl 17 . 2 (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
82, 4, 73eqtrd 2689 1 (𝜑 → (Edg‘𝐺) = {{𝑁}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {csn 4210  cop 4216  ran crn 5144  cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-edg 25985
This theorem is referenced by:  1loopgrnb0  26454  1loopgrvd2  26455
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