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 ) ) ) |
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 ) ) ) |