Theorem resabs1d 5922

Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
resabs1d.b	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
resabs1d	⊢ (𝜑 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))

Proof of Theorem resabs1d

Step	Hyp	Ref	Expression
1		resabs1d.b	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
2		resabs1 5921	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1539   ⊆ wss 3887   ↾ cres 5591

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595  df-rel 5596  df-res 5601

This theorem is referenced by:  f2ndf  7961  frrlem12  8113  ablfac1eulem  19675  kgencn2  22708  tsmsres  23295  resubmet  23965  xrge0gsumle  23996  cmssmscld  24514  cmsss  24515  cmscsscms  24537  minveclem3a  24591  dvmptresicc  25080  dvlip2  25159  c1liplem1  25160  efcvx  25608  logccv  25818  loglesqrt  25911  wilthlem2  26218  symgcom2  31353  cyc3conja  31424  bnj1280  33000  cvmlift2lem9  33273  nosupno  33906  nosupbnd1lem1  33911  nosupbnd2  33919  noinfno  33921  noinfbnd1lem1  33926  noinfbnd2  33934  mbfresfi  35823  ssbnd  35946  prdsbnd2  35953  cnpwstotbnd  35955  reheibor  35997  diophin  40594  fnwe2lem2  40876  dvsid  41949  limcresiooub  43183  limcresioolb  43184  fourierdlem46  43693  fourierdlem48  43695  fourierdlem49  43696  fourierdlem58  43705  fourierdlem72  43719  fourierdlem73  43720  fourierdlem74  43721  fourierdlem75  43722  fourierdlem89  43736  fourierdlem90  43737  fourierdlem91  43738  fourierdlem93  43740  fourierdlem100  43747  fourierdlem102  43749  fourierdlem103  43750  fourierdlem104  43751  fourierdlem107  43754  fourierdlem111  43758  fourierdlem112  43759  fourierdlem114  43761  afvres  44664  afv2res  44731  funcrngcsetc  45556  funcrngcsetcALT  45557  funcringcsetc  45593