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Theorem iedgvalsnop 26155
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 6336 . 2 (iEdg‘𝐺) = (iEdg‘{⟨𝐵, 𝐵⟩})
3 vtxvalsnop.b . . . 4 𝐵 ∈ V
43snopeqopsnid 5103 . . 3 {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
54fveq2i 6336 . 2 (iEdg‘{⟨𝐵, 𝐵⟩}) = (iEdg‘⟨{𝐵}, {𝐵}⟩)
6 snex 5037 . . 3 {𝐵} ∈ V
76, 6opiedgfvi 26111 . 2 (iEdg‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
82, 5, 73eqtri 2797 1 (iEdg‘𝐺) = {𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4317  cop 4323  cfv 6030  iEdgciedg 26096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fv 6038  df-2nd 7320  df-iedg 26098
This theorem is referenced by:  iedgval3sn  26157
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