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Theorem residm 5002
Description: Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
Assertion
Ref Expression
residm  |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )

Proof of Theorem residm
StepHypRef Expression
1 ssid 3210 . 2  |-  B  C_  B
2 resabs2 5001 . 2  |-  ( B 
C_  B  ->  (
( A  |`  B )  |`  B )  =  ( A  |`  B )
)
31, 2ax-mp 8 1  |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    C_ wss 3165    |` cres 4707
This theorem is referenced by:  resima  5003  dffv2  5608  fvsnun2  5732  qtopres  17405  eldioph2lem1  26942  eldioph2lem2  26943  bnj1253  29363
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-opab 4094  df-xp 4711  df-rel 4712  df-res 4717
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