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 ∪ 𝐵 ) ↔ ( Rel 𝐴 ∧ Rel 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unss | ⊢ ( ( 𝐴 ⊆ ( V × V ) ∧ 𝐵 ⊆ ( V × V ) ) ↔ ( 𝐴 ∪ 𝐵 ) ⊆ ( V × V ) ) | |
2 | df-rel | ⊢ ( Rel 𝐴 ↔ 𝐴 ⊆ ( V × V ) ) | |
3 | df-rel | ⊢ ( Rel 𝐵 ↔ 𝐵 ⊆ ( V × V ) ) | |
4 | 2 3 | anbi12i | ⊢ ( ( Rel 𝐴 ∧ Rel 𝐵 ) ↔ ( 𝐴 ⊆ ( V × V ) ∧ 𝐵 ⊆ ( V × V ) ) ) |
5 | df-rel | ⊢ ( Rel ( 𝐴 ∪ 𝐵 ) ↔ ( 𝐴 ∪ 𝐵 ) ⊆ ( V × V ) ) | |
6 | 1 4 5 | 3bitr4ri | ⊢ ( Rel ( 𝐴 ∪ 𝐵 ) ↔ ( Rel 𝐴 ∧ Rel 𝐵 ) ) |