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Theorem residm 5296
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 3516 . 2  |-  B  C_  B
2 resabs2 5295 . 2  |-  ( B 
C_  B  ->  (
( A  |`  B )  |`  B )  =  ( A  |`  B )
)
31, 2ax-mp 5 1  |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    C_ wss 3469    |` cres 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-opab 4499  df-xp 4998  df-rel 4999  df-res 5004
This theorem is referenced by:  resima  5297  dffv2  5931  fvsnun2  6088  qtopres  19927  eldioph2lem1  30284  eldioph2lem2  30285  icccncfext  31181  bnj1253  33027
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