Theorem usgredgop

Description: An edge of a simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017) (Proof shortened by Alexander van der Vekens, 16-Dec-2017) (Revised by AV, 15-Oct-2020)

	Ref	Expression
Assertion	usgredgop	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐸 = ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑋 ) = { 𝑀 , 𝑁 } ↔ ⟨ 𝑋 , { 𝑀 , 𝑁 } ⟩ ∈ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		usgrfun	⊢ ( 𝐺 ∈ USGraph → Fun ( iEdg ‘ 𝐺 ) )
2		funeq	⊢ ( 𝐸 = ( iEdg ‘ 𝐺 ) → ( Fun 𝐸 ↔ Fun ( iEdg ‘ 𝐺 ) ) )
3	1 2	syl5ibrcom	⊢ ( 𝐺 ∈ USGraph → ( 𝐸 = ( iEdg ‘ 𝐺 ) → Fun 𝐸 ) )
4	3	imp	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐸 = ( iEdg ‘ 𝐺 ) ) → Fun 𝐸 )
5		funopfvb	⊢ ( ( Fun 𝐸 ∧ 𝑋 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑋 ) = { 𝑀 , 𝑁 } ↔ ⟨ 𝑋 , { 𝑀 , 𝑁 } ⟩ ∈ 𝐸 ) )
6	4 5	stoic3	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐸 = ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐸 ) → ( ( 𝐸 ‘ 𝑋 ) = { 𝑀 , 𝑁 } ↔ ⟨ 𝑋 , { 𝑀 , 𝑁 } ⟩ ∈ 𝐸 ) )