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Theorem resabs1 5333
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 5315 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
2 sseqin2 3778 . . 3 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
3 reseq2 5298 . . 3 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
42, 3sylbi 205 . 2 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
51, 4syl5eq 2655 1 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  cin 3538  wss 3539  cres 5029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-xp 5033  df-rel 5034  df-res 5039
This theorem is referenced by:  resabs1d  5334  resabs2  5335  resiima  5385  fun2ssres  5830  fssres2  5969  smores3  7314  setsres  15677  gsum2dlem2  18141  lindsss  19929  resthauslem  20924  ptcmpfi  21373  tsmsres  21704  ressxms  22087  nrginvrcn  22253  xrge0gsumle  22391  lebnumii  22520  dfrelog  24060  relogf1o  24061  dvlog  24141  dvlog2  24143  efopnlem2  24147  wilthlem2  24539  gsumle  28903  rrhre  29186  iwrdsplit  29569  cvmsss2  30303  mbfposadd  32410  mzpcompact2lem  36115  eldioph2  36126  diophin  36137  diophrex  36140  2rexfrabdioph  36161  3rexfrabdioph  36162  4rexfrabdioph  36163  6rexfrabdioph  36164  7rexfrabdioph  36165  dvmptresicc  38592  fourierdlem46  38828  fourierdlem57  38839  fourierdlem111  38893  fouriersw  38907  psmeasurelem  39146
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