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Theorem relun 4709
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 3259 . 2  |-  ( ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V  X.  _V ) )  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
2 df-rel 4595 . . 3  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4595 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
42, 3anbi12i 681 . 2  |-  ( ( Rel  A  /\  Rel  B )  <->  ( A  C_  ( _V  X.  _V )  /\  B  C_  ( _V 
X.  _V ) ) )
5 df-rel 4595 . 2  |-  ( Rel  ( A  u.  B
)  <->  ( A  u.  B )  C_  ( _V  X.  _V ) )
61, 4, 53bitr4ri 271 1  |-  ( Rel  ( A  u.  B
)  <->  ( Rel  A  /\  Rel  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   _Vcvv 2727    u. cun 3076    C_ wss 3078    X. cxp 4578   Rel wrel 4585
This theorem is referenced by:  funun  5153  difxp  6005
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-in 3085  df-ss 3089  df-rel 4595
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