Theorem resabs1 6036

Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)

Assertion

Ref	Expression
resabs1	⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))

Proof of Theorem resabs1

Step	Hyp	Ref	Expression
1		resres 6022	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
2		sseqin2 4244	. . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵)
3		reseq2 6004	. . 3 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
4	2, 3	sylbi 217	. 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
5	1, 4	eqtrid 2792	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3975   ⊆ wss 3976   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707  df-res 5712

This theorem is referenced by:  resabs1d  6037  resabs2  6038  resiima  6105  fun2ssres  6623  fssres2  6789  smores3  8409  setsres  17225  gsum2dlem2  20013  lindsss  21867  resthauslem  23392  ptcmpfi  23842  tsmsres  24173  ressxms  24559  nrginvrcn  24734  xrge0gsumle  24874  lebnumii  25017  dvmptresicc  25971  dfrelog  26625  relogf1o  26626  dvlog  26711  dvlog2  26713  efopnlem2  26717  wilthlem2  27130  nosupres  27770  nosupbnd2lem1  27778  noinfres  27785  noinfbnd2lem1  27793  nosupinfsep  27795  gsumle  33074  rrhre  33967  iwrdsplit  34352  rpsqrtcn  34570  pthhashvtx  35095  cvmsss2  35242  mbfposadd  37627  mzpcompact2lem  42707  eldioph2  42718  diophin  42728  diophrex  42731  2rexfrabdioph  42752  3rexfrabdioph  42753  4rexfrabdioph  42754  6rexfrabdioph  42755  7rexfrabdioph  42756  resabs1i  45047  fourierdlem46  46073  fourierdlem57  46084  fourierdlem111  46138  fouriersw  46152  psmeasurelem  46391