Description: A class of ordered pairs is a relation. For a version without disjoint variable condition, but a longer proof using ax-13 , see relopabi . (Contributed by BJ, 22-Jul-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | relopabiv.1 | |- A = { <. x , y >. | ph } |
|
| Assertion | relopabiv | |- Rel A |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relopabiv.1 | |- A = { <. x , y >. | ph } |
|
| 2 | vex | |- x e. _V |
|
| 3 | vex | |- y e. _V |
|
| 4 | 2 3 | pm3.2i | |- ( x e. _V /\ y e. _V ) |
| 5 | 4 | a1i | |- ( ph -> ( x e. _V /\ y e. _V ) ) |
| 6 | 5 | ssopab2i | |- { <. x , y >. | ph } C_ { <. x , y >. | ( x e. _V /\ y e. _V ) } |
| 7 | df-xp | |- ( _V X. _V ) = { <. x , y >. | ( x e. _V /\ y e. _V ) } |
|
| 8 | 6 1 7 | 3sstr4i | |- A C_ ( _V X. _V ) |
| 9 | df-rel | |- ( Rel A <-> A C_ ( _V X. _V ) ) |
|
| 10 | 8 9 | mpbir | |- Rel A |