Theorem resabs1 6007

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 5993	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
2		sseqin2 4176	. . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵)
3		reseq2 5975	. . 3 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
4	2, 3	sylbi 220	. 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
5	1, 4	eqtrid 2812	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∩ cin 3905   ⊆ wss 3906   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176  df-xp 5669  df-rel 5670  df-res 5675

This theorem is used by:  resabs1i  6008  resabs1d  6009  resabs2  6010  resiima  6080  fun2ssres  6585  fssres2  6750  smores3  8342  setsres  17255  gsum2dlem2  20064  gsumle  20238  lindsss  22003  resthauslem  23549  ptcmpfi  23999  tsmsres  24330  ressxms  24711  nrginvrcn  24878  xrge0gsumle  25020  lebnumii  25154  dvmptresicc  26104  dfrelog  26759  relogf1o  26760  dvlog  26845  dvlog2  26847  efopnlem2  26851  wilthlem2  27262  nosupres  27900  nosupbnd2lem1  27908  noinfres  27915  noinfbnd2lem1  27923  nosupinfsep  27925  rrhre  34434  iwrdsplit  34801  rpsqrtcn  35004  pthhashvtx  35633  cvmsss2  35779  mbfposadd  38351  mzpcompact2lem  43515  eldioph2  43526  diophin  43536  diophrex  43539  2rexfrabdioph  43556  3rexfrabdioph  43557  4rexfrabdioph  43558  6rexfrabdioph  43559  7rexfrabdioph  43560  fourierdlem46  46899  fourierdlem57  46910  fourierdlem111  46964  fouriersw  46978  psmeasurelem  47217