Theorem vtxvalsnop 28926

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 6899	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘{⟨𝐵, 𝐵⟩})
3		vtxvalsnop.b	. . . 4 ⊢ 𝐵 ∈ V
4	3	snopeqopsnid 5511	. . 3 ⊢ {⟨𝐵, 𝐵⟩} = ⟨{𝐵}, {𝐵}⟩
5	4	fveq2i 6899	. 2 ⊢ (Vtx‘{⟨𝐵, 𝐵⟩}) = (Vtx‘⟨{𝐵}, {𝐵}⟩)
6		snex 5433	. . 3 ⊢ {𝐵} ∈ V
7	6, 6	opvtxfvi 28894	. 2 ⊢ (Vtx‘⟨{𝐵}, {𝐵}⟩) = {𝐵}
8	2, 5, 7	3eqtri 2757	1 ⊢ (Vtx‘𝐺) = {𝐵}

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3461  {csn 4630  ⟨cop 4636  ‘cfv 6549  Vtxcvtx 28881

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-1st 7994  df-vtx 28883

This theorem is referenced by:  vtxval3sn  28928