Theorem relun 4755

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 3321	. 2 |- ( ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) <-> ( A u. B ) C_ ( _V X. _V ) )
2		df-rel 4645	. . 3 |- ( Rel A <-> A C_ ( _V X. _V ) )
3		df-rel 4645	. . 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 4645	. 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 2749   u. cun 3139   C_ wss 3141   X. cxp 4636   Rel wrel 4643

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-rel 4645

This theorem is referenced by:  funun  5272