Theorem resabs1 5988

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 5974	. 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
2		sseqin2 4173	. . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵)
3		reseq2 5956	. . 3 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
4	2, 3	sylbi 219	. 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
5	1, 4	eqtrid 2808	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1559   ∩ cin 3901   ⊆ wss 3902   ↾ cres 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5160  df-xp 5649  df-rel 5650  df-res 5655

This theorem is referenced by:  resabs1i  5989  resabs1d  5990  resabs2  5991  resiima  6061  fun2ssres  6561  fssres2  6727  smores3  8318  setsres  17205  gsum2dlem2  20002  gsumle  20176  lindsss  21864  resthauslem  23411  ptcmpfi  23861  tsmsres  24192  ressxms  24573  nrginvrcn  24740  xrge0gsumle  24882  lebnumii  25016  dvmptresicc  25966  dfrelog  26618  relogf1o  26619  dvlog  26704  dvlog2  26706  efopnlem2  26710  wilthlem2  27121  nosupres  27759  nosupbnd2lem1  27767  noinfres  27774  noinfbnd2lem1  27782  nosupinfsep  27784  rrhre  34279  iwrdsplit  34645  rpsqrtcn  34848  pthhashvtx  35439  cvmsss2  35585  mbfposadd  38127  mzpcompact2lem  43293  eldioph2  43304  diophin  43314  diophrex  43317  2rexfrabdioph  43334  3rexfrabdioph  43335  4rexfrabdioph  43336  6rexfrabdioph  43337  7rexfrabdioph  43338  fourierdlem46  46687  fourierdlem57  46698  fourierdlem111  46752  fouriersw  46766  psmeasurelem  47005