Theorem relun 4472

Description: The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
relun	|- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) )

Proof of Theorem relun

Step	Hyp	Ref	Expression
1		unss 3146	. 2 |- ( ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) <-> ( A u. B ) C_ ( _V X. _V ) )
2		df-rel 4370	. . 3 |- ( Rel A <-> A C_ ( _V X. _V ) )
3		df-rel 4370	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
4	2, 3	anbi12i 447	. 2 |- ( ( Rel A /\ Rel B ) <-> ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) )
5		df-rel 4370	. 2 |- ( Rel ( A u. B ) <-> ( A u. B ) C_ ( _V X. _V ) )
6	1, 4, 5	3bitr4ri 211	1 |- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) )

Syntax hints:   /\ wa 102   <-> wb 103   _Vcvv 2601   u. cun 2971   C_ wss 2973   X. cxp 4361   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-rel 4370

This theorem is referenced by:  funun  4964