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Theorem residm 4664
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 3019 . 2  |-  B  C_  B
2 resabs2 4663 . 2  |-  ( B 
C_  B  ->  (
( A  |`  B )  |`  B )  =  ( A  |`  B )
)
31, 2ax-mp 7 1  |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    C_ wss 2974    |` cres 4367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-opab 3842  df-xp 4371  df-rel 4372  df-res 4377
This theorem is referenced by:  resima  4665  fvsnun2  5387
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