Theorem resabs1 5980

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 5966	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
2		sseqin2 4189	. . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵)
3		reseq2 5948	. . 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 2777	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3916   ⊆ wss 3917   ↾ cres 5643

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-xp 5647  df-rel 5648  df-res 5653

This theorem is referenced by:  resabs1i  5981  resabs1d  5982  resabs2  5983  resiima  6050  fun2ssres  6564  fssres2  6731  smores3  8325  setsres  17155  gsum2dlem2  19908  lindsss  21740  resthauslem  23257  ptcmpfi  23707  tsmsres  24038  ressxms  24420  nrginvrcn  24587  xrge0gsumle  24729  lebnumii  24872  dvmptresicc  25824  dfrelog  26481  relogf1o  26482  dvlog  26567  dvlog2  26569  efopnlem2  26573  wilthlem2  26986  nosupres  27626  nosupbnd2lem1  27634  noinfres  27641  noinfbnd2lem1  27649  nosupinfsep  27651  gsumle  33045  rrhre  34018  iwrdsplit  34385  rpsqrtcn  34591  pthhashvtx  35122  cvmsss2  35268  mbfposadd  37668  mzpcompact2lem  42746  eldioph2  42757  diophin  42767  diophrex  42770  2rexfrabdioph  42791  3rexfrabdioph  42792  4rexfrabdioph  42793  6rexfrabdioph  42794  7rexfrabdioph  42795  fourierdlem46  46157  fourierdlem57  46168  fourierdlem111  46222  fouriersw  46236  psmeasurelem  46475