MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resabs1 Unicode version

Theorem resabs1 5134
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 5118 . 2  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
2 sseqin2 3520 . . 3  |-  ( B 
C_  C  <->  ( C  i^i  B )  =  B )
3 reseq2 5100 . . 3  |-  ( ( C  i^i  B )  =  B  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  B )
)
42, 3sylbi 188 . 2  |-  ( B 
C_  C  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  B )
)
51, 4syl5eq 2448 1  |-  ( B 
C_  C  ->  (
( A  |`  C )  |`  B )  =  ( A  |`  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    i^i cin 3279    C_ wss 3280    |` cres 4839
This theorem is referenced by:  resabs2  5135  resiima  5179  fun2ssres  5453  fssres2  5570  f2ndf  6411  smores3  6574  tfrlem5  6600  setsres  13450  gsum2d  15501  ablfac1eulem  15585  resthauslem  17381  kgencn2  17542  ptcmpfi  17798  tsmsres  18126  ressxms  18508  nrginvrcn  18680  resubmet  18786  xrge0gsumle  18817  lebnumii  18944  cmsss  19256  minveclem3a  19281  dvlip2  19832  c1liplem1  19833  efcvx  20318  dfrelog  20416  relogf1o  20417  dvlog  20495  dvlog2  20497  efopnlem2  20501  logccv  20507  loglesqr  20595  wilthlem2  20805  rrhre  24340  cvmsss2  24914  cvmlift2lem9  24951  mbfresfi  26152  mbfposadd  26153  ssbnd  26387  prdsbnd2  26394  cnpwstotbnd  26396  reheibor  26438  mzpcompact2lem  26698  eldioph2  26710  diophin  26721  diophrex  26724  2rexfrabdioph  26746  3rexfrabdioph  26747  4rexfrabdioph  26748  6rexfrabdioph  26749  7rexfrabdioph  26750  fnwe2lem2  27016  lindsss  27162  dvsid  27416  afvres  27903  bnj1280  29095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-xp 4843  df-rel 4844  df-res 4849
  Copyright terms: Public domain W3C validator