MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relun Unicode version

Theorem relun 4982
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
StepHypRef Expression
1 unss 3513 . 2  |-  ( ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V  X.  _V ) )  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
2 df-rel 4876 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4876 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
42, 3anbi12i 679 . 2  |-  ( ( Rel  A  /\  Rel  B )  <->  ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V 
X.  _V ) ) )
5 df-rel 4876 . 2  |-  ( Rel  ( A  u.  B
)  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
61, 4, 53bitr4ri 270 1  |-  ( Rel  ( A  u.  B
)  <->  ( Rel  A  /\  Rel  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   _Vcvv 2948    u. cun 3310    C_ wss 3312    X. cxp 4867   Rel wrel 4874
This theorem is referenced by:  funun  5486  difxp  6371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-rel 4876
  Copyright terms: Public domain W3C validator