Theorem relun 4792

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 3347	. 2 |- ( ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) <-> ( A u. B ) C_ ( _V X. _V ) )
2		df-rel 4682	. . 3 |- ( Rel A <-> A C_ ( _V X. _V ) )
3		df-rel 4682	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
4	2, 3	anbi12i 460	. 2 |- ( ( Rel A /\ Rel B ) <-> ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) )
5		df-rel 4682	. 2 |- ( Rel ( A u. B ) <-> ( A u. B ) C_ ( _V X. _V ) )
6	1, 4, 5	3bitr4ri 213	1 |- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) )

Syntax hints:   /\ wa 104   <-> wb 105   _Vcvv 2772   u. cun 3164   C_ wss 3166   X. cxp 4673   Rel wrel 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-rel 4682

This theorem is referenced by:  funun  5315  fununfun  5317