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Theorem resabs1 5462
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 5444 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
2 sseqin2 3850 . . 3 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
3 reseq2 5423 . . 3 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
42, 3sylbi 207 . 2 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
51, 4syl5eq 2697 1 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  cin 3606  wss 3607  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150  df-res 5155
This theorem is referenced by:  resabs1d  5463  resabs2  5464  resiima  5515  fun2ssres  5969  fssres2  6110  smores3  7495  setsres  15948  gsum2dlem2  18416  lindsss  20211  resthauslem  21215  ptcmpfi  21664  tsmsres  21994  ressxms  22377  nrginvrcn  22543  xrge0gsumle  22683  lebnumii  22812  dfrelog  24357  relogf1o  24358  dvlog  24442  dvlog2  24444  efopnlem2  24448  wilthlem2  24840  gsumle  29907  rrhre  30193  iwrdsplit  30577  rpsqrtcn  30799  cvmsss2  31382  nosupres  31978  nosupbnd2lem1  31986  mbfposadd  33587  mzpcompact2lem  37631  eldioph2  37642  diophin  37653  diophrex  37656  2rexfrabdioph  37677  3rexfrabdioph  37678  4rexfrabdioph  37679  6rexfrabdioph  37680  7rexfrabdioph  37681  resabs1i  39650  dvmptresicc  40452  fourierdlem46  40687  fourierdlem57  40698  fourierdlem111  40752  fouriersw  40766  psmeasurelem  41005
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