Theorem vtxvalsnop

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.)

	Ref	Expression
Hypotheses	vtxvalsnop.b	⊢ 𝐵 ∈ V
	vtxvalsnop.g	⊢ 𝐺 = { ⟨ 𝐵 , 𝐵 ⟩ }
Assertion	vtxvalsnop	⊢ ( Vtx ‘ 𝐺 ) = { 𝐵 }

Proof

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