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Theorem 1loopgredg 26628
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 26161 . . 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 5508 . 2 (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
5 1loopgruspgr.a . . 3 (𝜑𝐴𝑋)
6 rnsnopg 5773 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
75, 6syl 17 . 2 (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
82, 4, 73eqtrd 2798 1 (𝜑 → (Edg‘𝐺) = {{𝑁}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {csn 4321  cop 4327  ran crn 5267  cfv 6049  Vtxcvtx 26094  iEdgciedg 26095  Edgcedg 26159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-edg 26160
This theorem is referenced by:  1loopgrnb0  26629  1loopgrvd2  26630
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