Theorem iedgvalsnop

Description: Degenerated case 2 for edges: The set of indexed edges 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	iedgvalsnop	⊢ ( iEdg ‘ 𝐺 ) = { 𝐵 }

Proof

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