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Theorem relun 4818
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 3362 . 2  |-  ( ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V  X.  _V ) )  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
2 df-rel 4712 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4712 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
42, 3anbi12i 678 . 2  |-  ( ( Rel  A  /\  Rel  B )  <->  ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V 
X.  _V ) ) )
5 df-rel 4712 . 2  |-  ( Rel  ( A  u.  B
)  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
61, 4, 53bitr4ri 269 1  |-  ( Rel  ( A  u.  B
)  <->  ( Rel  A  /\  Rel  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   _Vcvv 2801    u. cun 3163    C_ wss 3165    X. cxp 4703   Rel wrel 4710
This theorem is referenced by:  funun  5312  difxp  6169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-rel 4712
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