Description: Two ways of saying that a range Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020) (Revised by Peter Mazsa, 21-Sep-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | disjxrn | |- ( Disj ( R |X. S ) <-> ( ,~ `' R i^i ,~ `' S ) C_ _I ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrnrel | |- Rel ( R |X. S ) |
|
| 2 | dfdisjALTV2 | |- ( Disj ( R |X. S ) <-> ( ,~ `' ( R |X. S ) C_ _I /\ Rel ( R |X. S ) ) ) |
|
| 3 | 1 2 | mpbiran2 | |- ( Disj ( R |X. S ) <-> ,~ `' ( R |X. S ) C_ _I ) |
| 4 | 1cosscnvxrn | |- ,~ `' ( R |X. S ) = ( ,~ `' R i^i ,~ `' S ) |
|
| 5 | 4 | sseq1i | |- ( ,~ `' ( R |X. S ) C_ _I <-> ( ,~ `' R i^i ,~ `' S ) C_ _I ) |
| 6 | 3 5 | bitri | |- ( Disj ( R |X. S ) <-> ( ,~ `' R i^i ,~ `' S ) C_ _I ) |