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Theorem iedgvalsnop 29059
Description: Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.)
Hypotheses
Ref Expression
vtxvalsnop.b 𝐵 ∈ V
vtxvalsnop.g 𝐺 = {⟨𝐵, 𝐵⟩}
Assertion
Ref Expression
iedgvalsnop (iEdg‘𝐺) = {𝐵}

Proof of Theorem iedgvalsnop
StepHypRef Expression
1 vtxvalsnop.g . . 3 𝐺 = {⟨𝐵, 𝐵⟩}
21fveq2i 6909 . 2 (iEdg‘𝐺) = (iEdg‘{⟨𝐵, 𝐵⟩})
3 vtxvalsnop.b . . . 4 𝐵 ∈ V
43snopeqopsnid 5514 . . 3 {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
54fveq2i 6909 . 2 (iEdg‘{⟨𝐵, 𝐵⟩}) = (iEdg‘⟨{𝐵}, {𝐵}⟩)
6 snex 5436 . . 3 {𝐵} ∈ V
76, 6opiedgfvi 29027 . 2 (iEdg‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
82, 5, 73eqtri 2769 1 (iEdg‘𝐺) = {𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632  cfv 6561  iEdgciedg 29014
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
This theorem is referenced by:  iedgval3sn  29061
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