Theorem uhgredgiedgb

Description: In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020)

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

Proof

Step	Hyp	Ref	Expression
1		uhgredgiedgb.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2	1	uhgrfun	⊢ ( 𝐺 ∈ UHGraph → Fun 𝐼 )
3	1	edgiedgb	⊢ ( Fun 𝐼 → ( 𝐸 ∈ ( Edg ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ dom 𝐼 𝐸 = ( 𝐼 ‘ 𝑥 ) ) )
4	2 3	syl	⊢ ( 𝐺 ∈ UHGraph → ( 𝐸 ∈ ( Edg ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ dom 𝐼 𝐸 = ( 𝐼 ‘ 𝑥 ) ) )