Theorem resabs1 6012

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 5995	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
2		sseqin2 4216	. . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵)
3		reseq2 5977	. . 3 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
4	2, 3	sylbi 216	. 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
5	1, 4	eqtrid 2785	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∩ cin 3948   ⊆ wss 3949   ↾ cres 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684  df-res 5689

This theorem is referenced by:  resabs1d  6013  resabs2  6014  resiima  6076  fun2ssres  6594  fssres2  6760  smores3  8353  setsres  17111  gsum2dlem2  19839  lindsss  21379  resthauslem  22867  ptcmpfi  23317  tsmsres  23648  ressxms  24034  nrginvrcn  24209  xrge0gsumle  24349  lebnumii  24482  dvmptresicc  25433  dfrelog  26074  relogf1o  26075  dvlog  26159  dvlog2  26161  efopnlem2  26165  wilthlem2  26573  nosupres  27210  nosupbnd2lem1  27218  noinfres  27225  noinfbnd2lem1  27233  nosupinfsep  27235  gsumle  32242  rrhre  33001  iwrdsplit  33386  rpsqrtcn  33605  pthhashvtx  34118  cvmsss2  34265  mbfposadd  36535  mzpcompact2lem  41489  eldioph2  41500  diophin  41510  diophrex  41513  2rexfrabdioph  41534  3rexfrabdioph  41535  4rexfrabdioph  41536  6rexfrabdioph  41537  7rexfrabdioph  41538  resabs1i  43834  fourierdlem46  44868  fourierdlem57  44879  fourierdlem111  44933  fouriersw  44947  psmeasurelem  45186