Theorem resabs1d 5959

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 5957	. 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 1540   ⊆ wss 3903   ↾ cres 5621

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 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5155  df-xp 5625  df-rel 5626  df-res 5631

This theorem is referenced by:  f2ndf  8053  frrlem12  8230  ablfac1eulem  19953  funcrngcsetc  20525  funcrngcsetcALT  20526  funcringcsetc  20559  kgencn2  23442  tsmsres  24029  resubmet  24688  xrge0gsumle  24720  cmssmscld  25248  cmsss  25249  cmscsscms  25271  minveclem3a  25325  dvmptresicc  25815  dvlip2  25898  c1liplem1  25899  efcvx  26357  logccv  26570  loglesqrt  26669  wilthlem2  26977  nosupno  27613  nosupbnd1lem1  27618  nosupbnd2  27626  noinfno  27628  noinfbnd1lem1  27633  noinfbnd2  27641  symgcom2  33027  cyc3conja  33100  bnj1280  34993  cvmlift2lem9  35294  mbfresfi  37656  ssbnd  37778  prdsbnd2  37785  cnpwstotbnd  37787  reheibor  37829  diophin  42755  fnwe2lem2  43034  dvsid  44314  limcresiooub  45633  limcresioolb  45634  fourierdlem46  46143  fourierdlem48  46145  fourierdlem49  46146  fourierdlem58  46155  fourierdlem72  46169  fourierdlem73  46170  fourierdlem74  46171  fourierdlem75  46172  fourierdlem89  46186  fourierdlem90  46187  fourierdlem91  46188  fourierdlem93  46190  fourierdlem100  46197  fourierdlem102  46199  fourierdlem103  46200  fourierdlem104  46201  fourierdlem107  46204  fourierdlem111  46208  fourierdlem112  46209  fourierdlem114  46211  afvres  47166  afv2res  47233