Theorem resabs1d 5982

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 5980	. 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 3917   ↾ cres 5643

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-xp 5647  df-rel 5648  df-res 5653

This theorem is referenced by:  f2ndf  8102  frrlem12  8279  ablfac1eulem  20011  funcrngcsetc  20556  funcrngcsetcALT  20557  funcringcsetc  20590  kgencn2  23451  tsmsres  24038  resubmet  24697  xrge0gsumle  24729  cmssmscld  25257  cmsss  25258  cmscsscms  25280  minveclem3a  25334  dvmptresicc  25824  dvlip2  25907  c1liplem1  25908  efcvx  26366  logccv  26579  loglesqrt  26678  wilthlem2  26986  nosupno  27622  nosupbnd1lem1  27627  nosupbnd2  27635  noinfno  27637  noinfbnd1lem1  27642  noinfbnd2  27650  symgcom2  33048  cyc3conja  33121  bnj1280  35017  cvmlift2lem9  35305  mbfresfi  37667  ssbnd  37789  prdsbnd2  37796  cnpwstotbnd  37798  reheibor  37840  diophin  42767  fnwe2lem2  43047  dvsid  44327  limcresiooub  45647  limcresioolb  45648  fourierdlem46  46157  fourierdlem48  46159  fourierdlem49  46160  fourierdlem58  46169  fourierdlem72  46183  fourierdlem73  46184  fourierdlem74  46185  fourierdlem75  46186  fourierdlem89  46200  fourierdlem90  46201  fourierdlem91  46202  fourierdlem93  46204  fourierdlem100  46211  fourierdlem102  46213  fourierdlem103  46214  fourierdlem104  46215  fourierdlem107  46218  fourierdlem111  46222  fourierdlem112  46223  fourierdlem114  46225  afvres  47177  afv2res  47244