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Theorem iedgvalsnop 26842
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 6664 . 2 (iEdg‘𝐺) = (iEdg‘{⟨𝐵, 𝐵⟩})
3 vtxvalsnop.b . . . 4 𝐵 ∈ V
43snopeqopsnid 5386 . . 3 {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
54fveq2i 6664 . 2 (iEdg‘{⟨𝐵, 𝐵⟩}) = (iEdg‘⟨{𝐵}, {𝐵}⟩)
6 snex 5319 . . 3 {𝐵} ∈ V
76, 6opiedgfvi 26810 . 2 (iEdg‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
82, 5, 73eqtri 2851 1 (iEdg‘𝐺) = {𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2115  Vcvv 3480  {csn 4550  cop 4556  cfv 6343  iEdgciedg 26797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fv 6351  df-2nd 7685  df-iedg 26799
This theorem is referenced by:  iedgval3sn  26844
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