Metamath Proof Explorer

Theorem relun

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 ) )

Proof

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 ) )