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Theorem vtxvalsnop 26759
Description: Degenerated case 2 for vertices: The set of vertices 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
vtxvalsnop (Vtx‘𝐺) = {𝐵}

Proof of Theorem vtxvalsnop
StepHypRef Expression
1 vtxvalsnop.g . . 3 𝐺 = {⟨𝐵, 𝐵⟩}
21fveq2i 6672 . 2 (Vtx‘𝐺) = (Vtx‘{⟨𝐵, 𝐵⟩})
3 vtxvalsnop.b . . . 4 𝐵 ∈ V
43snopeqopsnid 5396 . . 3 {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
54fveq2i 6672 . 2 (Vtx‘{⟨𝐵, 𝐵⟩}) = (Vtx‘⟨{𝐵}, {𝐵}⟩)
6 snex 5328 . . 3 {𝐵} ∈ V
76, 6opvtxfvi 26727 . 2 (Vtx‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
82, 5, 73eqtri 2853 1 (Vtx‘𝐺) = {𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  Vcvv 3500  {csn 4564  cop 4570  cfv 6354  Vtxcvtx 26714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-1st 7685  df-vtx 26716
This theorem is referenced by:  vtxval3sn  26761
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