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Theorem relun 5107
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 3664 . 2  |-  ( ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V  X.  _V ) )  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
2 df-rel 4995 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4995 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
42, 3anbi12i 695 . 2  |-  ( ( Rel  A  /\  Rel  B )  <->  ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V 
X.  _V ) ) )
5 df-rel 4995 . 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 367   _Vcvv 3106    u. cun 3459    C_ wss 3461    X. cxp 4986   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-in 3468  df-ss 3475  df-rel 4995
This theorem is referenced by:  difxp  5416  funun  5612  fununfun  5614
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