Theorem resabs1d 5992

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 5990	. 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 1559   ⊆ wss 3904   ↾ cres 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-opab 5162  df-xp 5651  df-rel 5652  df-res 5657

This theorem is referenced by:  f2ndf  8094  frrlem12  8273  ablfac1eulem  20097  funcrngcsetc  20669  funcrngcsetcALT  20670  funcringcsetc  20703  kgencn2  23597  tsmsres  24184  resubmet  24842  xrge0gsumle  24874  cmssmscld  25392  cmsss  25393  cmscsscms  25415  minveclem3a  25469  dvmptresicc  25958  dvlip2  26037  c1liplem1  26038  efcvx  26489  logccv  26705  loglesqrt  26803  wilthlem2  27110  nosupno  27744  nosupbnd1lem1  27749  nosupbnd2  27757  noinfno  27759  noinfbnd1lem1  27764  noinfbnd2  27772  symgcom2  33225  cyc3conja  33298  bnj1280  35279  cvmlift2lem9  35625  mbfresfi  38129  ssbnd  38251  prdsbnd2  38258  cnpwstotbnd  38260  reheibor  38302  diophin  43317  fnwe2lem2  43592  dvsid  44871  limcresiooub  46180  limcresioolb  46181  fourierdlem46  46690  fourierdlem48  46692  fourierdlem49  46693  fourierdlem58  46702  fourierdlem72  46716  fourierdlem73  46717  fourierdlem74  46718  fourierdlem75  46719  fourierdlem89  46733  fourierdlem90  46734  fourierdlem91  46735  fourierdlem93  46737  fourierdlem100  46744  fourierdlem102  46746  fourierdlem103  46747  fourierdlem104  46748  fourierdlem107  46751  fourierdlem111  46755  fourierdlem112  46756  fourierdlem114  46758  afvres  47730  afv2res  47797