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Theorem uspgrloopedg 29536
Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29266) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
Hypothesis
Ref Expression
uspgrloopvtx.g 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
Assertion
Ref Expression
uspgrloopedg ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})

Proof of Theorem uspgrloopedg
StepHypRef Expression
1 uspgrloopvtx.g . . . 4 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
21fveq2i 6909 . . 3 (Edg‘𝐺) = (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3 snex 5436 . . . . 5 {⟨𝐴, {𝑁}⟩} ∈ V
43a1i 11 . . . 4 (𝐴𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5 edgopval 29068 . . . 4 ((𝑉𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
64, 5sylan2 593 . . 3 ((𝑉𝑊𝐴𝑋) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
72, 6eqtrid 2789 . 2 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
8 rnsnopg 6241 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
98adantl 481 . 2 ((𝑉𝑊𝐴𝑋) → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
107, 9eqtrd 2777 1 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632  ran crn 5686  cfv 6561  Edgcedg 29064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-2nd 8015  df-iedg 29016  df-edg 29065
This theorem is referenced by: (None)
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