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Theorem residm 5124
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 3460 . 2  |-  B  C_  B
2 resabs2 5123 . 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 1405    C_ wss 3413    |` cres 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-opab 4453  df-xp 4828  df-rel 4829  df-res 4834
This theorem is referenced by:  resima  5125  dffv2  5921  fvsnun2  6086  qtopres  20489  bnj1253  29387  eldioph2lem1  35034  eldioph2lem2  35035  relexpiidm  35663
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