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Theorem resabs1 5246
Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
resabs1  |-  ( B 
C_  C  ->  (
( A  |`  C )  |`  B )  =  ( A  |`  B )
)

Proof of Theorem resabs1
StepHypRef Expression
1 resres 5230 . 2  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
2 sseqin2 3676 . . 3  |-  ( B 
C_  C  <->  ( C  i^i  B )  =  B )
3 reseq2 5212 . . 3  |-  ( ( C  i^i  B )  =  B  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  B )
)
42, 3sylbi 195 . 2  |-  ( B 
C_  C  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  B )
)
51, 4syl5eq 2507 1  |-  ( B 
C_  C  ->  (
( A  |`  C )  |`  B )  =  ( A  |`  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    i^i cin 3434    C_ wss 3435    |` cres 4949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-opab 4458  df-xp 4953  df-rel 4954  df-res 4959
This theorem is referenced by:  resabs2  5247  resiima  5290  fun2ssres  5566  fssres2  5686  f2ndf  6787  smores3  6923  setsres  14319  gsum2dlem2  16583  gsum2dOLD  16585  ablfac1eulem  16694  lindsss  18377  resthauslem  19098  kgencn2  19261  ptcmpfi  19517  tsmsresOLD  19848  tsmsres  19849  ressxms  20231  nrginvrcn  20403  resubmet  20510  xrge0gsumle  20541  lebnumii  20669  cmsss  20992  minveclem3a  21045  dvlip2  21599  c1liplem1  21600  efcvx  22046  dfrelog  22149  relogf1o  22150  dvlog  22228  dvlog2  22230  efopnlem2  22234  logccv  22240  loglesqr  22328  wilthlem2  22539  gsumle  26390  rrhre  26591  iwrdsplit  26913  cvmsss2  27306  cvmlift2lem9  27343  mbfresfi  28585  mbfposadd  28586  ssbnd  28834  prdsbnd2  28841  cnpwstotbnd  28843  reheibor  28885  mzpcompact2lem  29235  eldioph2  29247  diophin  29258  diophrex  29261  2rexfrabdioph  29281  3rexfrabdioph  29282  4rexfrabdioph  29283  6rexfrabdioph  29284  7rexfrabdioph  29285  fnwe2lem2  29551  dvsid  29752  afvres  30225  bnj1280  32328
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