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Theorem res0 5856
Description: A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
Assertion
Ref Expression
res0 (𝐴 ↾ ∅) = ∅

Proof of Theorem res0
StepHypRef Expression
1 df-res 5566 . 2 (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V))
2 0xp 5648 . . 3 (∅ × V) = ∅
32ineq2i 4190 . 2 (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅)
4 in0 4349 . 2 (𝐴 ∩ ∅) = ∅
51, 3, 43eqtri 2853 1 (𝐴 ↾ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  Vcvv 3500  cin 3939  c0 4295   × cxp 5552  cres 5556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-opab 5126  df-xp 5560  df-res 5566
This theorem is referenced by:  ima0  5943  resdisj  6024  smo0  7986  tfrlem16  8020  tz7.44-1  8033  mapunen  8675  fnfi  8785  ackbij2lem3  9652  hashf1lem1  13803  setsid  16528  meet0  17737  join0  17738  frmdplusg  18002  psgn0fv0  18559  gsum2dlem2  19011  ablfac1eulem  19114  ablfac1eu  19115  psrplusg  20080  ply1plusgfvi  20329  ptuncnv  22334  ptcmpfi  22340  ust0  22746  xrge0gsumle  23359  xrge0tsms  23360  jensen  25483  egrsubgr  26976  0grsubgr  26977  pthdlem1  27464  0pth  27821  1pthdlem1  27831  eupth2lemb  27933  resf1o  30382  xrge0tsmsd  30609  gsumle  30642  esumsnf  31212  satfv1lem  32496  dfpo2  32878  eldm3  32884  rdgprc0  32925  zrdivrng  35102  eldioph4b  39276  diophren  39278  ismeannd  42618  psmeasure  42622  isomennd  42682  hoidmvlelem3  42748  aacllem  44737
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