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Theorem resabs1 4937
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 4921 . 2  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
2 sseqin2 3330 . . 3  |-  ( B 
C_  C  <->  ( C  i^i  B )  =  B )
3 reseq2 4903 . . 3  |-  ( ( C  i^i  B )  =  B  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  B )
)
42, 3sylbi 189 . 2  |-  ( B 
C_  C  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  B )
)
51, 4syl5eq 2300 1  |-  ( B 
C_  C  ->  (
( A  |`  C )  |`  B )  =  ( A  |`  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    i^i cin 3093    C_ wss 3094    |` cres 4628
This theorem is referenced by:  resabs2  4938  resiima  4982  fun2ssres  5198  fssres2  5312  fvres  5440  smores3  6303  tfrlem5  6329  setsres  13101  gsum2d  15150  ablfac1eulem  15234  resthauslem  17018  kgencn2  17179  ptcmpfi  17431  tsmsres  17753  ressxms  17998  nrginvrcn  18129  resubmet  18235  xrge0gsumle  18265  lebnumii  18391  cmsss  18699  minveclem3a  18718  dvlip2  19269  c1liplem1  19270  efcvx  19752  dfrelog  19850  relogf1o  19851  dvlog  19925  dvlog2  19927  efopnlem2  19931  logccv  19937  loglesqr  20025  wilthlem2  20234  cvmsss2  23142  cvmlift2lem9  23179  ssbnd  25844  prdsbnd2  25851  cnpwstotbnd  25853  reheibor  25895  mzpcompact2lem  26161  eldioph2  26173  diophin  26184  diophrex  26187  2rexfrabdioph  26209  3rexfrabdioph  26210  4rexfrabdioph  26211  6rexfrabdioph  26212  7rexfrabdioph  26213  fnwe2lem2  26480  lindsss  26626  dvsid  26880  bnj1280  28062
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4018  df-xp 4640  df-rel 4641  df-res 4646
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