Theorem vtxvalsnop 29014

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 6820	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘{⟨𝐵, 𝐵⟩})
3		vtxvalsnop.b	. . . 4 ⊢ 𝐵 ∈ V
4	3	snopeqopsnid 5444	. . 3 ⊢ {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
5	4	fveq2i 6820	. 2 ⊢ (Vtx‘{⟨𝐵, 𝐵⟩}) = (Vtx‘⟨{𝐵}, {𝐵}⟩)
6		snex 5369	. . 3 ⊢ {𝐵} ∈ V
7	6, 6	opvtxfvi 28982	. 2 ⊢ (Vtx‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
8	2, 5, 7	3eqtri 2758	1 ⊢ (Vtx‘𝐺) = {𝐵}

Syntax hints:   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {csn 4571  ⟨cop 4577  ‘cfv 6476  Vtxcvtx 28969

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fv 6484  df-1st 7916  df-vtx 28971

This theorem is referenced by:  vtxval3sn  29016