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Theorem 1loopgredg 27210
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 26761 . . 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 5801 . 2 (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
5 1loopgruspgr.a . . 3 (𝜑𝐴𝑋)
6 rnsnopg 6071 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
75, 6syl 17 . 2 (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
82, 4, 73eqtrd 2857 1 (𝜑 → (Edg‘𝐺) = {{𝑁}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {csn 4557  cop 4563  ran crn 5549  cfv 6348  Vtxcvtx 26708  iEdgciedg 26709  Edgcedg 26759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-edg 26760
This theorem is referenced by:  1loopgrnb0  27211  1loopgrvd2  27212
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