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Theorem resabs1 5876
Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
resabs1 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Proof of Theorem resabs1
StepHypRef Expression
1 resres 5859 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
2 sseqin2 4189 . . 3 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
3 reseq2 5841 . . 3 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
42, 3sylbi 218 . 2 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
51, 4syl5eq 2865 1 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  cin 3932  wss 3933  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554  df-rel 5555  df-res 5560
This theorem is referenced by:  resabs1d  5877  resabs2  5878  resiima  5937  fun2ssres  6392  fssres2  6539  smores3  7979  setsres  16513  gsum2dlem2  19020  lindsss  20896  resthauslem  21899  ptcmpfi  22349  tsmsres  22679  ressxms  23062  nrginvrcn  23228  xrge0gsumle  23368  lebnumii  23497  dfrelog  25076  relogf1o  25077  dvlog  25161  dvlog2  25163  efopnlem2  25167  wilthlem2  25573  gsumle  30652  rrhre  31161  iwrdsplit  31544  rpsqrtcn  31763  pthhashvtx  32271  cvmsss2  32418  nosupres  33104  nosupbnd2lem1  33112  mbfposadd  34820  mzpcompact2lem  39226  eldioph2  39237  diophin  39247  diophrex  39250  2rexfrabdioph  39271  3rexfrabdioph  39272  4rexfrabdioph  39273  6rexfrabdioph  39274  7rexfrabdioph  39275  resabs1i  41290  dvmptresicc  42080  fourierdlem46  42314  fourierdlem57  42325  fourierdlem111  42379  fouriersw  42393  psmeasurelem  42629
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