Theorem resabs1d 6012

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 6011	. 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 1541   ⊆ wss 3948   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682  df-rel 5683  df-res 5688

This theorem is referenced by:  f2ndf  8108  frrlem12  8284  ablfac1eulem  19983  kgencn2  23281  tsmsres  23868  resubmet  24538  xrge0gsumle  24569  cmssmscld  25091  cmsss  25092  cmscsscms  25114  minveclem3a  25168  dvmptresicc  25657  dvlip2  25736  c1liplem1  25737  efcvx  26185  logccv  26395  loglesqrt  26490  wilthlem2  26797  nosupno  27430  nosupbnd1lem1  27435  nosupbnd2  27443  noinfno  27445  noinfbnd1lem1  27450  noinfbnd2  27458  symgcom2  32503  cyc3conja  32574  bnj1280  34317  cvmlift2lem9  34588  mbfresfi  36837  ssbnd  36959  prdsbnd2  36966  cnpwstotbnd  36968  reheibor  37010  diophin  41812  fnwe2lem2  42095  dvsid  43392  limcresiooub  44657  limcresioolb  44658  fourierdlem46  45167  fourierdlem48  45169  fourierdlem49  45170  fourierdlem58  45179  fourierdlem72  45193  fourierdlem73  45194  fourierdlem74  45195  fourierdlem75  45196  fourierdlem89  45210  fourierdlem90  45211  fourierdlem91  45212  fourierdlem93  45214  fourierdlem100  45221  fourierdlem102  45223  fourierdlem103  45224  fourierdlem104  45225  fourierdlem107  45228  fourierdlem111  45232  fourierdlem112  45233  fourierdlem114  45235  afvres  46179  afv2res  46246  funcrngcsetc  46985  funcrngcsetcALT  46986  funcringcsetc  47022