Theorem resabs1d 5957

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 5955	. 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 3902   ↾ cres 5618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-opab 5154  df-xp 5622  df-rel 5623  df-res 5628

This theorem is referenced by:  f2ndf  8050  frrlem12  8227  ablfac1eulem  19987  funcrngcsetc  20556  funcrngcsetcALT  20557  funcringcsetc  20590  kgencn2  23473  tsmsres  24060  resubmet  24718  xrge0gsumle  24750  cmssmscld  25278  cmsss  25279  cmscsscms  25301  minveclem3a  25355  dvmptresicc  25845  dvlip2  25928  c1liplem1  25929  efcvx  26387  logccv  26600  loglesqrt  26699  wilthlem2  27007  nosupno  27643  nosupbnd1lem1  27648  nosupbnd2  27656  noinfno  27658  noinfbnd1lem1  27663  noinfbnd2  27671  symgcom2  33051  cyc3conja  33124  bnj1280  35030  cvmlift2lem9  35353  mbfresfi  37712  ssbnd  37834  prdsbnd2  37841  cnpwstotbnd  37843  reheibor  37885  diophin  42811  fnwe2lem2  43090  dvsid  44370  limcresiooub  45686  limcresioolb  45687  fourierdlem46  46196  fourierdlem48  46198  fourierdlem49  46199  fourierdlem58  46208  fourierdlem72  46222  fourierdlem73  46223  fourierdlem74  46224  fourierdlem75  46225  fourierdlem89  46239  fourierdlem90  46240  fourierdlem91  46241  fourierdlem93  46243  fourierdlem100  46250  fourierdlem102  46252  fourierdlem103  46253  fourierdlem104  46254  fourierdlem107  46257  fourierdlem111  46261  fourierdlem112  46262  fourierdlem114  46264  afvres  47209  afv2res  47276