Theorem resabs1d 5348

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 5347	. 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 1475   ⊆ wss 3540   ↾ cres 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-res 5050

This theorem is referenced by:  f2ndf  7170  ablfac1eulem  18294  kgencn2  21170  tsmsres  21757  resubmet  22413  xrge0gsumle  22444  cmsss  22955  minveclem3a  23006  dvlip2  23562  c1liplem1  23563  efcvx  24007  logccv  24209  loglesqrt  24299  wilthlem2  24595  bnj1280  30342  cvmlift2lem9  30547  mbfresfi  32626  ssbnd  32757  prdsbnd2  32764  cnpwstotbnd  32766  reheibor  32808  diophin  36354  fnwe2lem2  36639  dvsid  37552  limcresiooub  38709  limcresioolb  38710  dvmptresicc  38809  fourierdlem46  39045  fourierdlem48  39047  fourierdlem49  39048  fourierdlem58  39057  fourierdlem72  39071  fourierdlem73  39072  fourierdlem74  39073  fourierdlem75  39074  fourierdlem89  39088  fourierdlem90  39089  fourierdlem91  39090  fourierdlem93  39092  fourierdlem100  39099  fourierdlem102  39101  fourierdlem103  39102  fourierdlem104  39103  fourierdlem107  39106  fourierdlem111  39110  fourierdlem112  39111  fourierdlem114  39113  afvres  39901  funcrngcsetc  41790  funcrngcsetcALT  41791  funcringcsetc  41827