Description: Alternate definition of relation: a class is a relation if and only if it is equal to its "relation part" (see relpart ). (Contributed by NM, 14-May-2008) Do not depend on df-cnv to move it earlier. (Revised by BJ, 16-Oct-2026)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | bj-dfrel3 | ⊢ ( Rel 𝑅 ↔ ( 𝑅 ↾ V ) = 𝑅 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfss2 | ⊢ ( 𝑅 ⊆ ( V × V ) ↔ ( 𝑅 ∩ ( V × V ) ) = 𝑅 ) | |
| 2 | df-rel | ⊢ ( Rel 𝑅 ↔ 𝑅 ⊆ ( V × V ) ) | |
| 3 | df-res | ⊢ ( 𝑅 ↾ V ) = ( 𝑅 ∩ ( V × V ) ) | |
| 4 | 3 | eqeq1i | ⊢ ( ( 𝑅 ↾ V ) = 𝑅 ↔ ( 𝑅 ∩ ( V × V ) ) = 𝑅 ) |
| 5 | 1 2 4 | 3bitr4i | ⊢ ( Rel 𝑅 ↔ ( 𝑅 ↾ V ) = 𝑅 ) |