Description: The union of two relations is a relation. Compare Exercise 5 of TakeutiZaring p. 25. (Contributed by NM, 12-Aug-1994)
Ref | Expression | ||
---|---|---|---|
Assertion | relun | |- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unss | |- ( ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) <-> ( A u. B ) C_ ( _V X. _V ) ) |
|
2 | df-rel | |- ( Rel A <-> A C_ ( _V X. _V ) ) |
|
3 | df-rel | |- ( Rel B <-> B C_ ( _V X. _V ) ) |
|
4 | 2 3 | anbi12i | |- ( ( Rel A /\ Rel B ) <-> ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) ) |
5 | df-rel | |- ( Rel ( A u. B ) <-> ( A u. B ) C_ ( _V X. _V ) ) |
|
6 | 1 4 5 | 3bitr4ri | |- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) ) |