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Theorem resabs1d 5877
Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
resabs1d.b (𝜑𝐵𝐶)
Assertion
Ref Expression
resabs1d (𝜑 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Proof of Theorem resabs1d
StepHypRef Expression
1 resabs1d.b . 2 (𝜑𝐵𝐶)
2 resabs1 5876 . 2 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
31, 2syl 17 1 (𝜑 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wss 3933  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554  df-rel 5555  df-res 5560
This theorem is referenced by:  f2ndf  7805  ablfac1eulem  19123  kgencn2  22093  tsmsres  22679  resubmet  23337  xrge0gsumle  23368  cmssmscld  23880  cmsss  23881  cmscsscms  23903  minveclem3a  23957  dvlip2  24519  c1liplem1  24520  efcvx  24964  logccv  25173  loglesqrt  25266  wilthlem2  25573  symgcom2  30655  cyc3conja  30726  bnj1280  32189  cvmlift2lem9  32455  frrlem12  33031  nosupno  33100  nosupbnd1lem1  33105  nosupbnd2  33113  mbfresfi  34819  ssbnd  34947  prdsbnd2  34954  cnpwstotbnd  34956  reheibor  34998  diophin  39247  fnwe2lem2  39529  dvsid  40540  limcresiooub  41799  limcresioolb  41800  dvmptresicc  42080  fourierdlem46  42314  fourierdlem48  42316  fourierdlem49  42317  fourierdlem58  42326  fourierdlem72  42340  fourierdlem73  42341  fourierdlem74  42342  fourierdlem75  42343  fourierdlem89  42357  fourierdlem90  42358  fourierdlem91  42359  fourierdlem93  42361  fourierdlem100  42368  fourierdlem102  42370  fourierdlem103  42371  fourierdlem104  42372  fourierdlem107  42375  fourierdlem111  42379  fourierdlem112  42380  fourierdlem114  42382  afvres  43248  afv2res  43315  funcrngcsetc  44197  funcrngcsetcALT  44198  funcringcsetc  44234
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