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

Step	Hyp	Ref	Expression
1		resres 5118	. 2 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
2		sseqin2 3564	. . 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 ) )
4	2, 3	sylbi 195	. 2 |- ( B C_ C -> ( A |` ( C i^i B ) ) = ( A |` B ) )
5	1, 4	syl5eq 2482	1 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1369   i^i cin 3322   C_ wss 3323   |` cres 4837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-opab 4346  df-xp 4841  df-rel 4842  df-res 4847

This theorem is referenced by:  resabs2  5135  resiima  5178  fun2ssres  5454  fssres2  5574  f2ndf  6673  smores3  6806  setsres  14194  gsum2dlem2  16450  gsum2dOLD  16452  ablfac1eulem  16561  lindsss  18228  resthauslem  18942  kgencn2  19105  ptcmpfi  19361  tsmsresOLD  19692  tsmsres  19693  ressxms  20075  nrginvrcn  20247  resubmet  20354  xrge0gsumle  20385  lebnumii  20513  cmsss  20836  minveclem3a  20889  dvlip2  21442  c1liplem1  21443  efcvx  21889  dfrelog  21992  relogf1o  21993  dvlog  22071  dvlog2  22073  efopnlem2  22077  logccv  22083  loglesqr  22171  wilthlem2  22382  gsumle  26197  rrhre  26399  iwrdsplit  26722  cvmsss2  27115  cvmlift2lem9  27152  mbfresfi  28391  mbfposadd  28392  ssbnd  28640  prdsbnd2  28647  cnpwstotbnd  28649  reheibor  28691  mzpcompact2lem  29041  eldioph2  29053  diophin  29064  diophrex  29067  2rexfrabdioph  29087  3rexfrabdioph  29088  4rexfrabdioph  29089  6rexfrabdioph  29090  7rexfrabdioph  29091  fnwe2lem2  29357  dvsid  29558  afvres  30031  bnj1280  31898