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Theorem resid2 25200
Description: Any operation can be restricted to  ( _V  X.  _V ). (Contributed by FL, 20-Mar-2011.)
Assertion
Ref Expression
resid2  |-  ( ( Rel  A  /\  Rel  dom 
A )  ->  ( A  |`  ( _V  X.  _V ) )  =  A )

Proof of Theorem resid2
StepHypRef Expression
1 df-rel 4712 . 2  |-  ( Rel 
dom  A  <->  dom  A  C_  ( _V  X.  _V ) )
2 relssres 5008 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  ->  ( A  |`  ( _V  X.  _V ) )  =  A )
31, 2sylan2b 461 1  |-  ( ( Rel  A  /\  Rel  dom 
A )  ->  ( A  |`  ( _V  X.  _V ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   _Vcvv 2801    C_ wss 3165    X. cxp 4703   dom cdm 4705    |` cres 4707   Rel wrel 4710
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dm 4715  df-res 4717
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