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Theorem res0 5946
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 5650 . 2 (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V))
2 0xp 5735 . . 3 (∅ × V) = ∅
32ineq2i 4174 . 2 (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅)
4 in0 4356 . 2 (𝐴 ∩ ∅) = ∅
51, 3, 43eqtri 2769 1 (𝐴 ↾ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3448  cin 3914  c0 4287   × cxp 5636  cres 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-xp 5644  df-res 5650
This theorem is referenced by:  ima0  6034  resdisj  6126  dfpo2  6253  smo0  8309  tfrlem16  8344  tz7.44-1  8357  rdg0n  8385  mapunen  9097  fnfi  9132  ackbij2lem3  10184  hashf1lem1  14360  hashf1lem1OLD  14361  setsid  17087  join0  18301  meet0  18302  frmdplusg  18671  psgn0fv0  19300  gsum2dlem2  19755  ablfac1eulem  19858  ablfac1eu  19859  psrplusg  21365  ply1plusgfvi  21629  ptuncnv  23174  ptcmpfi  23180  ust0  23587  xrge0gsumle  24212  xrge0tsms  24213  jensen  26354  egrsubgr  28267  0grsubgr  28268  pthdlem1  28756  0pth  29111  1pthdlem1  29121  eupth2lemb  29223  fressupp  31645  resf1o  31689  xrge0tsmsd  31941  gsumle  31974  zarcmplem  32502  esumsnf  32703  satfv1lem  33996  eldm3  34373  rdgprc0  34407  bj-rdg0gALT  35571  zrdivrng  36441  disjresin  36726  eldioph4b  41163  diophren  41165  ismeannd  44782  psmeasure  44786  isomennd  44846  hoidmvlelem3  44912  aacllem  47322
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