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Theorem relun 5057
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 3631 . 2  |-  ( ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V  X.  _V ) )  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
2 df-rel 4948 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4948 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
42, 3anbi12i 697 . 2  |-  ( ( Rel  A  /\  Rel  B )  <->  ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V 
X.  _V ) ) )
5 df-rel 4948 . 2  |-  ( Rel  ( A  u.  B
)  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
61, 4, 53bitr4ri 278 1  |-  ( Rel  ( A  u.  B
)  <->  ( Rel  A  /\  Rel  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   _Vcvv 3071    u. cun 3427    C_ wss 3429    X. cxp 4939   Rel wrel 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073  df-un 3434  df-in 3436  df-ss 3443  df-rel 4948
This theorem is referenced by:  difxp  5363  funun  5561  fununfun  5563
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