Description: The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a Cartesian square is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | trinxp | |- ( ( R o. R ) C_ R -> ( ( R i^i ( A X. A ) ) o. ( R i^i ( A X. A ) ) ) C_ ( R i^i ( A X. A ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xpidtr | |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A ) |
|
| 2 | trin2 | |- ( ( ( R o. R ) C_ R /\ ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A ) ) -> ( ( R i^i ( A X. A ) ) o. ( R i^i ( A X. A ) ) ) C_ ( R i^i ( A X. A ) ) ) |
|
| 3 | 1 2 | mpan2 | |- ( ( R o. R ) C_ R -> ( ( R i^i ( A X. A ) ) o. ( R i^i ( A X. A ) ) ) C_ ( R i^i ( A X. A ) ) ) |