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

Theorem resabs1 5961
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 5947 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
2 sseqin2 4176 . . 3 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
3 reseq2 5929 . . 3 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
42, 3sylbi 217 . 2 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
51, 4eqtrid 2776 1 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3904  wss 3905  cres 5625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5158  df-xp 5629  df-rel 5630  df-res 5635
This theorem is referenced by:  resabs1i  5962  resabs1d  5963  resabs2  5964  resiima  6031  fun2ssres  6531  fssres2  6696  smores3  8283  setsres  17108  gsum2dlem2  19869  gsumle  20043  lindsss  21750  resthauslem  23267  ptcmpfi  23717  tsmsres  24048  ressxms  24430  nrginvrcn  24597  xrge0gsumle  24739  lebnumii  24882  dvmptresicc  25834  dfrelog  26491  relogf1o  26492  dvlog  26577  dvlog2  26579  efopnlem2  26583  wilthlem2  26996  nosupres  27636  nosupbnd2lem1  27644  noinfres  27651  noinfbnd2lem1  27659  nosupinfsep  27661  rrhre  34007  iwrdsplit  34374  rpsqrtcn  34580  pthhashvtx  35120  cvmsss2  35266  mbfposadd  37666  mzpcompact2lem  42744  eldioph2  42755  diophin  42765  diophrex  42768  2rexfrabdioph  42789  3rexfrabdioph  42790  4rexfrabdioph  42791  6rexfrabdioph  42792  7rexfrabdioph  42793  fourierdlem46  46153  fourierdlem57  46164  fourierdlem111  46218  fouriersw  46232  psmeasurelem  46471
  Copyright terms: Public domain W3C validator