Theorem vtxvalsnop 29040

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

Step	Hyp	Ref	Expression
1		vtxvalsnop.g	. . 3 ⊢ 𝐺 = {⟨𝐵, 𝐵⟩}
2	1	fveq2i 6834	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘{⟨𝐵, 𝐵⟩})
3		vtxvalsnop.b	. . . 4 ⊢ 𝐵 ∈ V
4	3	snopeqopsnid 5454	. . 3 ⊢ {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
5	4	fveq2i 6834	. 2 ⊢ (Vtx‘{⟨𝐵, 𝐵⟩}) = (Vtx‘⟨{𝐵}, {𝐵}⟩)
6		snex 5378	. . 3 ⊢ {𝐵} ∈ V
7	6, 6	opvtxfvi 29008	. 2 ⊢ (Vtx‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
8	2, 5, 7	3eqtri 2760	1 ⊢ (Vtx‘𝐺) = {𝐵}

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3437  {csn 4577  ⟨cop 4583  ‘cfv 6489  Vtxcvtx 28995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fv 6497  df-1st 7930  df-vtx 28997

This theorem is referenced by:  vtxval3sn  29042