Theorem edgiedgb

Description: A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020) (Revised by AV, 8-Dec-2021)

	Ref	Expression
Hypothesis	edgiedgb.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	edgiedgb	⊢ ( Fun 𝐼 → ( 𝐸 ∈ ( Edg ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ dom 𝐼 𝐸 = ( 𝐼 ‘ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		edgiedgb.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2		edgval	⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
3	1	eqcomi	⊢ ( iEdg ‘ 𝐺 ) = 𝐼
4	3	rneqi	⊢ ran ( iEdg ‘ 𝐺 ) = ran 𝐼
5	2 4	eqtri	⊢ ( Edg ‘ 𝐺 ) = ran 𝐼
6	5	eleq2i	⊢ ( 𝐸 ∈ ( Edg ‘ 𝐺 ) ↔ 𝐸 ∈ ran 𝐼 )
7		elrnrexdmb	⊢ ( Fun 𝐼 → ( 𝐸 ∈ ran 𝐼 ↔ ∃ 𝑥 ∈ dom 𝐼 𝐸 = ( 𝐼 ‘ 𝑥 ) ) )
8	6 7	syl5bb	⊢ ( Fun 𝐼 → ( 𝐸 ∈ ( Edg ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ dom 𝐼 𝐸 = ( 𝐼 ‘ 𝑥 ) ) )